关于连通可分拓扑空间上的可测效用概念：一个序同构定理
On the notion of measurable utility on a connected and separable topological space: an order isomorphism theorem
摘要：
本文的目的是定义基数效用函数的概念，称为可测效用，并将其定义在弱序拓扑空间的连通可分子集上。这个定义与Frisch在1926年和Shapley在1975年给出的定义是等价的，并且假设了一组备选方案的公理，这些公理允许对备选方案进行顺序排序并比较它们的效用差异。在简要回顾了功利主义的哲学思想和效用理论的历史之后，本文介绍了表示效用强度比较的数学框架，并证明了一系列拓扑引理，这些引理将用于主要结果。最后，本文给出并证明了定义在弱序拓扑空间的连通可分子集上的可测效用函数的一个表示定理，该拓扑空间在其乘积上具有另一个弱序。在对序关系的某些假设下，主要定理证明了在正仿射变换下，与实直线的序同构的存在性和唯一性。
英文摘要：
The aim of this article is to define a notion of cardinal utility function called measurable utility and to define it on a connected and separable subset of a weakly ordered topological space. The definition is equivalent to the ones given by Frisch in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that allow both to ordinally rank alternatives and to compare their utility differences. After a brief review of the philosophy of utilitarianism and the history of utility theory, the paper introduces the mathematical framework to represent intensity comparisons of utility and proves a list of topological lemmas that will be used in the main result. Finally, the article states and proves a representation theorem for a measurable utility function defined on a connected and separable subset of a weakly ordered topological space equipped with another weak order on its cartesian product. Under some assumptions on the order relations, the main theorem proves existence and uniqueness, up to positive affine transformations, of an order isomorphism with the real line.
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