作用空间可数的模糊对策及其在广义拟变量不等式组中的应用

摘要

我们引入了一个具有可数作用空间的抽象模糊经济（广义模糊对策）模型，并研究了模糊均衡的存在性。作为应用，我们证明了本文定义的具有随机模糊映射的广义拟变量不等式组解的存在性。我们的结果通过考虑值为完备可数度量空间上模糊集的随机模糊映射，为当前的文献带来了新的内容。

1引言

由Borglin和Keiting形式化的经典抽象经济（或广义博弈）模型[1]或者Shafer和Sonnenschein[2]由一组有限的代理组成，每个代理都有特定的约束和偏好，并通过通信进行描述。这个模型在阿罗和德布鲁所考虑的经济研究中发挥了核心作用[三]在过去的几十年中，它在许多方面得到了推广。Kim和Lee[4]证明了扎德提出的模糊集理论[5]当约束或偏好由于代理人的行为而不精确时，已经成为研究广义博弈均衡存在性的一个很好的框架。这些作者证明了与此主题相关的第一个定理。对于较新的结果，读者应该参考，例如[6–8].

本文的目的是双重的。首先，它扩展了作者定义的带有私人信息和可数行为集的抽象经济模型的研究（参见[9])在模糊设置中，通过考虑代理可能对其选择的操作具有部分控制权的情况。特别注意模糊均衡的存在性。根据Huang的定义，借助模糊随机映射描述了选举情况下代理人的个人特征所导致的不确定性[10]. 其次，应用关于具有可数动作集的模糊对策的模糊均衡存在性的定理，证明了具有随机模糊映射的广义拟均衡不等式组解的存在性。目前的工作受到了该领域正在进行的研究的激励和启发。我们提到张和朱[11]是第一个引入模糊映射变分不等式概念的人。此后，人们研究了确定性和模糊性环境中的几类变分不等式。例如，我们引用Noor[12]、Park和Jeong[13]丁和帕克[14]. 最近已经考虑并研究了随机变分不等式（例如，参见[15–19]). 本文的结果，包括在非模糊设置中得出的结果，是全新的，以前没有文献报道过。

论文的其余部分组织如下。下一节给出了一些符号和术语约定，然后第3.1节介绍了带有私有信息和可计数行动空间的抽象模糊经济模型。主要结果涉及第3.2节中所述的模糊平衡的存在性。第4.1节定义了具有随机模糊映射的随机拟变分不等式的新类型系统。第4.2节证明了它们解的存在性。第4.3节包含非模糊设置中的结果，显示为先前获得的定理的结果。最后，第5节给出了我们的研究结论。

2符号和定义

Zadeh介绍的模糊集理论[5]，是一个在数学经济学中非常常用的框架。模糊集的特征是其任何元素的隶属度都可以是单位区间内的任意数$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$相反，只有二进制对$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\}">\}</trans>$可以给候选元素赋予清晰集的隶属度。

我们在下面介绍一些与此理论相关的概念。

如果Y（Y）是拓扑空间，然后是函数A类从Y（Y）进入之内$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$被称为上的模糊集Y（Y）（见Chang[20]). 上的所有模糊集族Y（Y）表示为$\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$.如果X（X）和Y（Y）是拓扑空间，然后是映射$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$称为模糊映射.如果P（P）是一个模糊映射，那么对于每个$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$,$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$是上的模糊集Y（Y）和$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$,$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}$被称为隶属度 年在里面$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$.让$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$，然后是集合${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\}">\}</trans>$称为坚强的 一-割集模糊集的A类.

为了方便读者，我们复习了一些关于对应关系可测性的基本定义和结果。

让$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$成为一个可测量的空间，Y（Y）是拓扑空间，并且$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$是一封信件。F类是弱 可测量的如果${\mathrm{<trans\; data-src="F">F类</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="T">T型</trans>}$对于每个开放子集A类属于Y（Y）（下逆${\mathrm{<trans\; data-src="F">F类</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}$，也称为弱逆，属于子集A类属于Y（Y）由定义${\mathrm{<trans\; data-src="F">F类</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\cap ">\cap </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans>$).F类是可测量的如果${\mathrm{<trans\; data-src="F">F类</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="T">T型</trans>}$对于每个封闭子集A类属于Y（Y）.如果$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$是一个可测量的空间，Y（Y）是一个可数集合$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$那么是一封信件F类是可测量的，如果$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}$,${\mathrm{<trans\; data-src="F">F类</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$是$\mathcal{<trans\; data-src="T">T型</trans>}$-可衡量的。对于通信$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$从一个可测空间到一个可度量空间，如果F类是可测量的，那么它也是弱可测量的并且如果F类是紧值且弱可测的，它是可测的。可以从不同方面找到对应关系的可测量性，例如[21–24].

已经定义了随机模糊映射，以便对生成不精确值数据的随机机制进行建模，这些不精确值数据可以通过使用模糊集来适当地描述。

让Y（Y）是一个拓扑空间，$\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$是所有模糊集的集合Y（Y）和$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src=")">)</trans>$成为一个可测量的空间。模糊映射$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$据说是可测量的（请参见[25])如果有任何给定$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$,${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$是一个可测集值映射。模糊映射$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$据说有一个可测量的图表如果有任何给定$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$，集值映射${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$有一个可测量的图表。模糊映射$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$称为随机模糊映射如果，对于任何给定$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$,$\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$是一个可度量的模糊映射。

3具有私有信息和可数行动集的抽象模糊经济的模糊均衡存在性

数学经济学和博弈论的一个新方向涉及这样一个事实，即在不同的经济活动中，表征代理人决策个体特征的不确定性必须包括在数学模型中。不确定性可以用随机模糊映射来描述。因此，模糊性框架已经成为应用数学语言的一部分。

本节致力于定义一个新的具有私有信息和可数行为集的抽象模糊经济模型，并证明模糊均衡的存在性。该模型是Borglin和Keiting提出的抽象经济经典确定性模型的扩展[1]或者Shafer和Sonnenschein[2]. 他们只考虑了有限的代理集。该模型概括了[9]通过考虑模糊设置。它不同于文献中存在私有信息的其他抽象模糊经济模型（参见[26]或[27])由于随机模糊约束映射和随机模糊偏好映射的值是可数完备度量空间上的模糊集，因此使用了对应分布理论。作者介绍的基于可测量性要求的其他模型是[7,28,29].

3.1带有私人信息的抽象模糊经济模型

抽象经济是一个中间模型，它将纳什类型的一般经济联系起来，这些一般经济描述了一般的竞争情况，而不是具体的（参见[30])Arrow-Debreu模型属于经济领域，因此是一种特殊模型（参见[三]).

我们定义了一个新的抽象模糊经济模型，该模型包含私有信息和可计数的行为集。

定义1具有私有信息和可数行动空间的抽象模糊经济（或广义模糊博弈）定义如下：

$$\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

哪里我是非空有限集（代理集），并且：

• （a） ${\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是agent的动作（策略）模糊映射我;

• （b） ${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是随机模糊映射（agent的约束映射我);

• （c） ${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是随机模糊映射（agent的偏好映射我);

• （d） ${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$是随机模糊约束函数${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$是随机模糊偏好函数；

• （e） ${\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$是这样的$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\subset ">\subset </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$和${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\subset ">\subset </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$.

现在，我们将解释模型的要素，并给出解释。

我是一个非空的有限集（代理集）。对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$、行动空间${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是可数完全度量空间，并且$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是一个可测量的空间。$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src=")">)</trans>$产品是可测量的空间吗$<trans\; data-src="(">(</trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{<trans\; data-src="⨂">⨂</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$、和μ是上的概率测度$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src=")">)</trans>$。就这点而言$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$，我们定义坐标投影${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.随机映射${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans>$被解释为玩家我的私人信息与他的行为有关。

我们还表示，对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$,$<trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$可测映射集$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$到${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.一个元素${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$属于$<trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$称为纯战略对于玩家我.A型纯战略概况 克是一个n个-向量函数$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$它为每个参与者指定了一个纯策略。

我们假设存在一个模糊映射${\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$这样每个代理我可以从中选择操作${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\subset ">\subset </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.功能${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$据说是一种选择${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$如果${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

${\mathrm{<trans\; data-src="D">D类</trans>}}_{{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}}$是布景吗$<trans\; data-src="\{">\{</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\text{<trans data-src=" is a measurable selection of ">是一个可衡量的选择</trans>}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\}">\}</trans>$和${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathcal{<trans\; data-src="D">D类</trans>}}_{{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}}$.

对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$，我们表示${\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$，其中${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是一个可测量的选择${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$和${\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src=")">)</trans>$.

对于每个代理我使用随机模糊映射描述约束和偏好${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$分别地${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.在世界上$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$，数字${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$关联到$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$可以解释为强度年优先于${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$或其真实程度年优先于${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$。我们还可以看到其价值${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$，与关联$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$，作为球员的信念我在这个州${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$他可以选择$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}$.元素${\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是世界上每个状态下的动作水平，${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans>$表达战略可行性的感知程度克和${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans>$表示策略的首选级别克.

下面介绍了该模型的模糊平衡概念。它概括了Shafer和Sonnenschein对平衡的确定定义[2]以及作者在[9].

定义2Γ的模糊平衡被定义为策略剖面${\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$这样，对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• (1) ${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$;

• (2) ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

3.2具有可数作用集的抽象模糊经济的模糊均衡的存在性

本小节旨在建立抽象模糊经济的模糊均衡的存在性。这些假设涉及定义模型的对应关系的可测性和上半连续性。

以下定义和性质是用来证明模糊均衡存在性的基本工具。

让X（X）,Y（Y）是拓扑空间$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$是一封信件。F类据说是上半连续如果每个$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}$和每个开放集V（V）在里面Y（Y）具有$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="V">V（V）</trans>}$，有一个开放的社区U型属于x个在里面X（X）这样的话$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="V">V（V）</trans>}$对于每个$\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="U">U型</trans>}$.

引理1[19]

让 X（X） 和 Y（Y） 是两个拓扑空间且设 A类 是的一个开放子集 X（X）.假设 ${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$,${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 上半连续 ${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$ 对所有人来说 $\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}$.然后是信件 $\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 由定义

$$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>& \mathit{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\notin ">\notin </trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\\ {\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>& \mathit{\text{<trans data-src="if ">如果</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="A">A类</trans>}\end{array}$$

也是上半连续的.

我们跟随于和张[31]. 让Y（Y）是一个可数完全度量空间，$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$是一个无原子的概率空间$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$是一种可衡量的通信。功能$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}$据说是精选的 F类如果$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$对于λ-几乎$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="T">T型</trans>}$让我们表示${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="F">F类</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="f">（f）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="f">（f）</trans>}\text{<trans data-src=" is a measurable selection of ">是一个可测量的选择</trans>}\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="\}">\}</trans>$.

我们将介绍${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="F">F类</trans>}}$也是于和张获得的[31]. 下一个引理状态——的凸性和紧性${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="F">F类</trans>}}$用于任何通信F类.

引理2[31]

让 Y（Y） 是可数完全度量空间,$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$ 是一个无原子的概率空间 $\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 是可衡量的通信.然后 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="F">F类</trans>}}$ 不是-空间中的空和凸 $\mathcal{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$-概率测度的空间 Y（Y）,具有弱收敛拓扑.

引理3[31]

让 Y（Y） 是可数完全度量空间,$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$ 是一个无原子的概率空间 $\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 是可衡量的对应关系.如果 F类 是紧值的,然后 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="F">F类</trans>}}$ 是紧凑的 $\mathcal{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=")">)</trans>$.

引理4[31]

让 X（X） 是度量空间,$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="T">T型</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$ 是一个无原子的概率空间,Y（Y） 是可数完备度量空间 $\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 是一封信件.假设对于任何固定的 x个 在里面 X（X）,$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$(也表示为 ${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="x">x个</trans>}}$)是紧凑型-有值可测对应,对于每个固定的 $\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="T">T型</trans>}$,$\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>$ 上半连续 X（X）.阿尔索,假设存在一个紧集-有价值的通信 $\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}$ 这样的话 $\mathrm{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>\mathrm{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$ 对所有人来说 t吨 和 x个.然后 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="x">x个</trans>}}}$ 上半连续 X（X）.

我们还需要Kuratowski-Ryll-Nardzewski选择定理来证明我们的主要结果。

定理1（Kuratowski-Ryll-Nardzewski选择定理）[32]

与非-从可测空间到波兰空间的空闭值允许一个可测选择器.

让我们表示$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

定理2是本文的主要结果。它将定理3推广到[9]通过考虑模糊框架。我们强调，我们对模糊均衡存在性的论证与文献中使用的方法不同，因为私人信息和可数行为空间的建模采用了新的设置。该证明尤其依赖于Kuratowski-Ryll-Nardzewski选择定理和Fan不动点定理。这一结果的可能应用可能涉及经济或市场中可计数方面的问题。

定理2 让 $\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=")">)</trans>$ 是一种抽象的模糊经济,拥有私人信息和可计数的行动空间,哪里 μ 是无原子的 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是紧值的;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 可测量且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值;

• （c） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是可衡量的并且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值;

• （d） 对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$ 和每个 $\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="g">克</trans>}}<trans\; data-src=")">)</trans>}$;

• （e） 成套设备 ${\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans>$ 在中打开 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

然后就有了 ${\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$ 平衡Γ.

证明可以有效地利用不动点方法。为此，我们将构建几个通信。

让$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$固定。

让我们表示

$$\begin{array}{c}{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans>\text{<trans data-src="and">和</trans>}\hfill \\ {\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=".">.</trans>\hfill \end{array}$$

让我们定义${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$通过

$$\begin{array}{c}{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\begin{array}{ll}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}& \text{<trans data-src="if ">如果</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans>{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\\ {<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}& \text{<trans data-src="if ">如果</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\end{array}\text{<trans data-src="and">和</trans>}\hfill \\ \mathrm{<trans\; data-src="\Phi ">\Phi </trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\Phi ">\Phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{<trans\; data-src="\prod ">\prod </trans>}{\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>\hfill \end{array}$$

对于每个$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$，其中${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\text{<trans data-src=" is a measurable selection of ">是一个可测量的选择</trans>}{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$.

我们将Ky-Fan不动点定理应用于对应Φ，并将得到一个不动点的存在性，它将是抽象经济Γ的平衡点。为此，我们检查相关集合的属性和对应关系${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$和Φ。

首先，我们注意到${\mathrm{<trans\; data-src="D">D类</trans>}}_{{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}}$根据引理2，它是非空的和凸的，根据引理3，它是紧的。因此，集合${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$也是非空的、紧凑的和凸的。

根据假设（b）和（c）${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$具有非空且紧凑的值，并且可以测量${\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$假设（e）意味着集合${\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$在中打开${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$假设（b）和（c）意味着${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="z">z（z）</trans>}}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是上半连续的；因此，我们可以应用引理1来断言${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$相对于上半连续$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

此外，对于每个$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$非空、凸且紧凑。每个人的非空性${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$隐含在通信中存在可测量的选择${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$根据Kuratowski-Ryll-Nardzewski选择定理。引理2和引理3也保证了集合的凸性和紧性${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$，其中$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

根据引理4，对应关系${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是上半连续的。那么对应Φ是上半连续的，并且具有非空值、紧值和凸值。我们还证明了它定义在非空的凸紧集上。我们可以应用Ky Fan不动点定理[33]到Φ，我们得到存在一个不动点${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Phi ">\Phi </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$。特别是，对于每个玩家$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$从定义${\mathcal{<trans\; data-src="D">D类</trans>}}_{{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$，我们得出结论，对于每个玩家我，存在${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="Meas">Meas公司</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$这样的话${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$是一个选择${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$和${\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$让我们表示${\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>$.

我们证明${\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$是Γ的平衡。对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$，因为${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}$是一个选择${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>$，因此${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$如果$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans>{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$或${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$如果$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

根据假设（d），可以得出如下结论${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.然后${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$和$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$这相当于以下事实${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}$和${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{{\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}$对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$。因此，${\mathrm{<trans\; data-src="g">克</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$是Γ的平衡。□

4个随机模糊映射的随机拟变量不等式

证明抽象经济均衡存在性的技术通常用于解决与此领域相关的其他问题，特别是变分不等式、极小极大定理、一些类拟均衡问题以及交换经济均衡的存在性。可以进一步获得第3节结果的一些应用。本文主要讨论了一类新的拟变量不等式组的定义及其解的存在性证明，其中随机模糊映射的值是完备可数度量空间上的模糊集。

变分不等式理论考虑了在数学、物理、经济和工程科学的各个领域中使用的大量有趣和重要的工具。

4.1新型广义拟变量不等式组

受张朱以来发展起来的大型文学的启发[11]引入了模糊映射的变分不等式的概念，自Noor和Elsanousi以来[17]定义了随机变分不等式的概念，我们还提出了一类新的广义拟变分不等式系统，在这方面似乎很受欢迎。

我们将使用第3节中的相同设置。

对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$，让${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是一个可数完全度量空间。让${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是一个模糊映射，并且让${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$是一个模糊函数。让${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$.

现在，我们介绍一类新的广义拟变量不等式组，如下所示。

• (1) 查找${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$以及所有人${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$,

哪里${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{{\mathrm{<trans\; data-src="i">我</trans>}}_{{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}}<trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="Y">Y（Y）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{{\mathrm{<trans\; data-src="i">我</trans>}}_{{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$.

我们注意到（1）推广了几位作者所考虑的拟变量不等式的一种特殊确定性情形(例如见袁[19]以及其中的参考文献）和Tan和Yuan在[18]或元（参见[34]以及其中的参考）。

如果${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是一封经典的信件${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$作为（1）的特例，我们得到了如下变分不等式。

（2） 查找${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$以及所有人${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

（i）${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$;

（ii）${<trans\; data-src="sup">啜饮</trans>}_{{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

最后，我们定义了以下广义拟变量不等式系统，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$,${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是一个可数的完全可度量拓扑向量空间，${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}$是的双重空间${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>$是模糊映射和${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$,${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$是模糊函数。

• (3) 查找${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$以及所有人${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="〉">〉</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$,

其中配对的真实部分${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}$和${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$表示为$<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="〉">〉</trans>$对于每个$\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}$和$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

如果${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$和${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}}$是经典的对应关系，然后我们得到以下变分不等式。

• (4) 查找${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$以及所有人${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="〉">〉</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

系统（4）推广了几位作者考虑的拟变量不等式的一个特殊确定性情形(例如见袁[35]以及其中的参考文献）和袁研究的随机拟变量不等式（参见[19]以及其中的参考）。

我们的拟变分不等式系统与文献中已有的拟变分不等式系统在以下几个方面不同：考虑了随机模糊映射，并且它们的值是完全可数度量空间上的模糊集。

4.2具有随机模糊映射的广义拟变量不等式组解的存在性

在本小节中，我们建立了关于具有随机模糊映射的广义随机拟变量不等式组的存在性的新结果。这些证明主要基于抽象模糊经济的模糊均衡存在定理。

为了获得下一小节中的定理，我们在这里设置了以下通用条件。

让我是一个非空的有限集。对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$,${\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是可数完全度量空间，并且$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是一个可测量的空间。让$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="Z">Z</trans>}<trans\; data-src=")">)</trans>$成为产品可测量的空间$<trans\; data-src="(">(</trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{<trans\; data-src="⨂">⨂</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$然后让μ是关于的无原子概率测度$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="Z">Z</trans>}<trans\; data-src=")">)</trans>$对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$，信件${\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是可衡量的。

这是我们的第一个定理。

定理3 假设每个都满足以下条件 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 具有紧凑值;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 可测量且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值.

让我们假设映射 ${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$ 是这样的:

• （c） $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续，紧致值为 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 对于每个固定 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$;

• （d） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 对于每个固定 $<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$;

• （e） 对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 在中弱打开 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,哪里 ${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}$ 由定义

  $${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathit{\text{<trans data-src="for each ">对于每个</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=";">;</trans>$$

• （f） $<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$ 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 这样，对于每一个 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$ 并且对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

证明对于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$，让${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$和${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$这样的话${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$对于每个$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

我们将展示抽象经济$\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=")">)</trans>$满足定理2的所有假设。

为此，让我们考虑${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.

根据（c），我们有$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\to ">\to </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是上半连续的，具有非空值，根据（d），${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}$对于每个$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

根据的定义${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$，我们注意到，对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$以便$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="\varnothing ">\varnothing </trans>}<trans\; data-src="\}">\}</trans>$在中弱打开${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$由（e）签署。

根据（b）和（f），对于每个$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$，信件${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$和${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是可衡量的。

因此，抽象模糊经济$\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}<trans\; data-src=")">)</trans>$满足定理2的所有假设。因此，存在${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$并且对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

$$\begin{array}{c}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}\text{<trans data-src="and">和</trans>}\hfill \\ {<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\cap ">\cap </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src=";">;</trans>\hfill \end{array}$$

也就是说，存在${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$并且对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

 □

备注1上述定理可以与[34]它说明了在以下情况下随机拟变量不等式解的存在性结果${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{<trans\; data-src="\prod ">\prod </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}}{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$每个都是上半连续的$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$和$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$.

如果$<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，我们得到以下推论。

推论1 让 S公司 是可数完全度量空间,$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="Z">Z</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans>$ 成为测量空间,哪里 μ 是无原子的.假设满足以下条件:

• （a） 信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 具有紧凑值;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 是可衡量的并且,对所有人来说 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$,信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 与非上半连续-空压缩值.

映射 $\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$ 是这样的:

• （c） $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 上半连续，紧致值为 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 对于每个固定 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$;

• （d） $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 对于每个固定 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\times ">\times </trans>$ ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$;

• （e） 对于每个 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$,$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 在中弱打开 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,哪里 $\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="Z">Z</trans>}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}$ 由定义

  $$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>\mathit{\text{<trans data-src="for each ">对于每个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Z">Z</trans>}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=";">;</trans>$$

• （f） $<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="Z">Z</trans>}$ 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 这样，每一个人 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

作为定理3的结果，我们证明了以下Tan和Yuan的类型[18]具有随机模糊映射的广义随机拟变分不等式系统。

定理4 假设每个都满足以下条件 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 具有紧凑值;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是可衡量的并且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值.

${\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="F">F类</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src=")">)</trans>$ 和 ${\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$ 是这样的:

• （c） 对于每个固定 $<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 具有紧值的上半连续;

• （d） 对于每个固定 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,成套设备 $<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 在中弱打开 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$;

• （e） $<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$ 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 这样，对于每一个 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$ 并且对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 对所有人来说 $\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$.

证明让我们定义${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$通过

$${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans>\text{<trans data-src="for each ">对于每个</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=".">.</trans>$$

我们有${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$对于每个固定$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$.

满足定理3的所有假设。根据定理3，存在${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$这样，对于每一个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$并且对于每个${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$:

$$\begin{array}{c}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}\text{<trans data-src="and">和</trans>}\hfill \\ \underset{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}\underset{\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="g">克</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>}}{<trans\; data-src="sup">啜饮</trans>}<trans\; data-src="Re">重新</trans><trans\; data-src="〈">〈</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="〉">〉</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

与推论1类似，如果$<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$. □

以下定理是作为定理3的一个特例获得的。

定理5 假设每个都满足以下条件 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 具有紧凑值;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是可衡量的并且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,信件 ${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值.

然后存在 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 这样每 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$ 并且对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$.

如果$<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，我们得到以下结果。

定理6 让 S公司 是一个可数完备度量空间 $<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="Z">Z</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src=")">)</trans>$ 成为测量空间,哪里 μ 是无原子的.假设满足以下条件:

• （a） 信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 具有紧凑值;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$,信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 是可衡量的并且,对所有人来说 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$,信件 ${<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$ 上半连续-空压缩值.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}}$ 这样的话,并且对于每个 $\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}$,${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}$.

4.3广义拟变量不等式的经典系统

作为最后一小节中所述定理的结果，获得了关于非模糊集合中广义随机拟变量不等式组的几个结果。它们是新的，尚未在文献中报道。我们在第4.2节开始时设定的相同条件下工作。我们提出定理7作为定理3的结果。

定理7 假设每个都满足以下条件 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 可测量且具有紧凑值;

• （b） 信件 ${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$ 是这样的，对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$,${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是可衡量的并且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值.

让我们假设映射 ${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}<trans\; data-src="\cup ">\cup </trans><trans\; data-src="\{">\{</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="\}">\}</trans>$ 是这样的:

• （c） $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续，上有紧致值 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$ 对于每个固定 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$;

• （d） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\notin ">\notin </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 对于每个固定 $<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$;

• （e） 对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$ 在中弱打开 ${\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$,哪里 ${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="R">R（右）</trans>}$ 由定义 ${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$ 对于每个 $<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$;

• （f） $<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="Z">Z</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$ 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$ 这样，对于每一个 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$ 并且对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,

• （i） ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$;

• （ii） ${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

下一个定理涉及完全可数度量空间中值的对应。

定理8 假设每个都满足以下条件 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$:

• （a） 信件 ${\mathrm{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 用紧凑值进行测量;

• （b） 对于每个 $\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$,${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 是可衡量的并且,对所有人来说 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdot ">\cdot </trans><trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="2">2</trans>}}^{{\mathrm{<trans\; data-src="S">S公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$ 上半连续-空压缩值.

然后就有了 ${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="X">X（X）</trans>}}$ 这样，对于每一个 $\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="I">我</trans>}$ 并且对于每个 ${\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$,${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\omega ">\omega </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="\ast ">\ast </trans>}<trans\; data-src=")">)</trans>$.

5结束语

本文的主要目的是定义一个具有私有信息和可数行为集的抽象模糊经济的新模型，并证明在上半连续假设下模糊均衡的存在性。用模糊随机映射描述了选举中代理人的个体特征所产生的不确定性。利用证明平衡点存在性的技巧，解决了一类新的带有随机模糊映射的拟变量不等式组，该类不等式组在一个特殊章节中被引入。

需要进一步关注已建立结果在模糊经济领域和模糊博弈论中的应用研究。本文的方法和设置可以适用于求解其他类型的广义变分不等式或拟平衡问题，特别适用于一般平衡理论。