概率测度函子的开多交换性

关于幂等概率测度函子的完全度量化

On perfect metrizability of the functor of idempotent probability measures

本文证明了作用在范畴内的幂等概率测度的函子及其连续映射是完全度量化的。

幂等概率测度，I

Idempotent probability measures, I

弱 * 拓扑确定的紧Hausdorff空间上的所有幂等概率测度（Maslov测度）的集合在紧T2空间的范畴$\comp$上是函子的.我们证明了所得到的函子在E.谢平此外，这个函子是$\comp$上monad的函子部分。证明了幂等概率测度单子包含超空间单子作为其子单子。本文对幂等概率测度定义了一个对应的Milyon-map概念。利用Milyanov映射存在的事实，证明了幂等概率测度的函子保持开满映射类.与概率测度的情况不同，分配给因子上的每对幂等概率测度的对应关系是不开放的。

关于精确容量

On exact capacities

本文将容量视为单调合作规范游戏，引入拓扑学中的精确游戏类，并证明其为容量 functor 的子functor，介于凸容量和平衡容量之间，同时探讨概率测度作为游戏核心元素的表示及精确容量的开放性。

Geometrical properties of space $I_f(X)$ of idempotent probability measures

In the paper we proved that for a compact $X$ inclusion $I_{f}(X)\in ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor $I_{f}$ preserves property of a compact to be $Q$-manyfold or a Hilbert cube, preserves property of map layers to be $ANR$-compact, compact $Q$-manyfold and a Hilbert cube (the finite sum of Hilbert cubes)

Geometrical properties of the space $P_f(X)$ of probability measures

In this paper we prove that for a compact space $X$ inclusion $P_{f}(X)\in ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor $P_{f}$ preserves property of a compact to be $Q$-manifold or a Hilbert cube, properties of maps fibres to be $ANR$-compact, $Q$-manifold, Hilbert cube (the finite of Hilbert cube).

概率测度空间的拓扑，II

Topology of probability measure spaces, II

本文是作者“概率测度空间的拓扑，I”一文的继续，致力于研究概率$τ$ -光滑空间和Radon测度空间的函子$\hat P$和$P_τ$.在这一部分中，我们研究了Radon概率测度空间的重心映射。应用所得结果证明了函子$\hat P$在可度量化空间范畴中是一元的.证明了函子$\hat P$和$P_τ$允许提升到有界度量空间范畴$BMetr$和一致空间范畴$Unif$，并研究了这些提升的性质.

The topology of spaces of probability measures, I

For a Tychonoff space $X$, the constructions $\hat P(X)$ and $P_τ(X)$ of the spaces of probability Radon measures and probability $τ$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in the category of Tychonoff spaces, which extend the functor $P$ of probability measures in the category of compacta. In this part we investigate general topological properties of the spaces $\hat P(X)$ and $P_τ(X)$, as well as categorial properties of the functors $\hat P$ and $P_τ$.

度量空间之间的概率测度与Mily-marts映射

Probability measures and Milyutin maps between metric spaces

证明了Radon概率测度的函子$\Hat{P}$将完全度量化空间之间的开映射变换为软映射。这个结果被应用于建立完全度量化空间之间Milyerkmann映射的一些性质。

嵌入卷积的紧Clifford半群的刻画与函子半群

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

研究了拓扑群上概率测度卷积半群的代数和拓扑性质，证明了紧Clifford拓扑半群$S$嵌入到某个拓扑群$G$上的卷积半群$P(G)$中当且仅当$S$嵌入到$G$的紧子集半群$\exp(G)$中当且仅当$S$是逆半群且具有零维极大半格.我们还证明了这样一个Clifford半群$S$嵌入函子半群$F(G)$中，对于每个弱正规一元函子$F$，在一个适当的紧拓扑群$G$上，使$F(G)$包含一个$G$ -不变元（它是$G$上Haar测度的模拟）。

超度量空间上的极大极小测度

Max-min measures on ultrametric spaces

超度量空间上紧支撑的所有概率测度的集合的超度量化首先由Hartog和de Vink定义。本文考虑超度量空间上的极大极小测度的一个类似构造。特别地，我们证明了函子的极大极小测度与幂等测度是同构的。然而，我们表明，这不是由这些函子生成的单子的情况。

Complete Quasi-Metrics for Hyperspaces, Continuous Valuations, and Previsions

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi-metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda-complete, and that the associated topology is the weak topology. We do this not only for probability distributions, …

Measure and integration on Boolean algebras of regular open subsets in a topological space

The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. We then explain the relationship between credences, residual charges, and Borel probability measures. We show that a credence can be represented by a normal Borel measure, augmented with a `liminal structure', which specifies how two or more regular open sets share the pro…

一种关于普适可测性的新方法

A New Approach to Universal Measurability

本文针对概率测度空间嵌入问题给出否定答案，转而研究交替测度空间，证明其满足相应的C-嵌入性质，并推广了普适可测性概念。