A New Approach to Universal Measurability
DOI:
https://doi.org/10.58649/axcznr61Keywords:
measure spaces, probability measures, Radon measures, universally measurable sets, universal measure zero setsAbstract
V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that Pτ(X) is C-embedded in Pσ(X); (ii) is it true that PR(X) is C-embedded in PR(βX) if and only if X is pseudocompact, where Pσ, Pτ, and PR 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 Mσ, Mτ, and MR, the situation changes. It is proved that (i) Mτ(X) is C-embedded in Mσ(X); (ii) MR(X) is C-embedded in MR(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.
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