AI 中文总结
本文从拓扑视角研究Mislove定义的随机变量，证明归一化√-max简单随机变量空间构成T0空间范畴的单子，归一化√-max连续随机变量空间构成d空间范畴的单子，还涉及清醒空间上的相关性质。
AI 中文摘要
Mislove、Goubault和Varacca研究了如何在论域理论中定义随机变量，以构成有界完备论域范畴上的单子，他们旨在用这类随机变量单子对概率程序设计语言进行建模。本文从拓扑视角聚焦于Mislove定义的随机变量，为T0空间上的√-max连续随机变量提供了一种拓扑，构造了一个新的T0空间，其中√-max性质对单子结构至关重要。我们证明，归一化√-max简单随机变量的空间构成T0空间范畴上的单子，而归一化√-max连续随机变量的空间则给出d空间范畴上的单子。此外，在清醒空间上，归一化√-max连续随机变量的空间是归一化√-max简单随机变量空间的清醒化。
英文摘要
Mislove, Goubault and Varacca investigated how to define random variables in Domain theory to form monads over the category of bounded complete domains. They intended to model probabilistic programming languages with their random variables monads. In this paper, we focus on the random variables defined by Mislove from a topological perspective. We provide a topology for $\surd$-max continuous random variables on a $T_0$ space, we construct a new $T_0$ space, where $\surd$-max property is essential for the monad structures. We show that the spaces of normalized $\surd$-max simple random variables form a monad over the category of $T_0$ spaces and that the spaces of normalized $\surd$-max continuous random variables give a monad over the category of d-spaces. In addition, on a sober space, the space of normalized $\surd$-max continuous random variables is the sobrification of the space of normalized $\surd$-max simple random variables.
