一种关于普适可测性的新方法
A New Approach to Universal Measurability
- Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
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。
AI中文摘要：
V. Fedorchuk、A. Chizogidze 和 T. Banakh 于 2003 年以及 V. Bogachev 于 2024 年提出了以下问题：(i) $P_\tau(X)$ 是否 $C$-嵌入于 $P_\sigma(X)$；(ii) $P_R(X)$ 是否 $C$-嵌入于 $P_R(\beta X)$ 当且仅当 $X$ 是伪紧的，其中 $P_\sigma$、$P_\tau$ 和 $P_R$ 分别是空间 $X$ 上关于 Baire $\sigma$-代数的概率 $\sigma$-可加、$\tau$-可加以及 Radon 测度的函子。这些问题的答案是否定的。然而，如果我们考虑相应的交替测度 $M_\sigma$、$M_\tau$ 和 $M_R$ 来代替概率测度，情况则发生变化。我们证明了 (i) $M_\tau(X)$ 是 $C$-嵌入于 $M_\sigma(X)$ 的；(ii) $M_R(X)$ 是 $C$-嵌入于 $M_R(\beta X)$ 的当且仅当 $X$ 是伪紧的。测度空间的 $C$-嵌入问题是测度空间重合问题的推广，而后者是普适可测集和普适测度零集经典概念的发展。我们获得了一个一般性定理，该定理导出了上述结果。
英文摘要：
V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_τ(X)$ is $C$-embedded in $P_σ(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(βX)$ if and only if $X$ is pseudocompact, where $P_σ$, $P_τ$, and $P_R$ are the functors of probability $σ$-additive on the Baire $σ$-algebra, $τ$-additive, and Radon measures on the space $X$? The answers to these questions are negative. However, if instead of probability measures we consider the corresponding alternating measures $M_σ$, $M_τ$, and $M_R$, the situation changes. It is proved that (i) $M_τ(X)$ is $C$-embedded in $M_σ(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(βX)$ if and only if $X$ is pseudocompact. The question of $C$-embedding of measure spaces is an extension of the question of coincidence of measure spaces, which is a development of the classical concepts of universally measurable and universal measure zero sets. A general theorem is obtained, which leads to the mentioned results.
