A New Approach to Universal Measurability

V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_{\tau }(X)$ is $C$-embedded in $P_{\sigma }(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(\beta X)$ if and only if $X$ is pseudocompact, where $P_{\sigma }$, $P_{\tau }$, and $P_R$ are the functors of probability $\sigma$-additive on the Baire $\sigma$-algebra, $\tau$-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 $M_{\sigma }$, $M_{\tau }$, and $M_R$, the situation changes. It is proved that (i) $M_{\tau }(X)$ is $C$-embedded in $M_{\sigma }(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(\beta X)$ 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.