A New Approach to Universal Measurability
Abstract
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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BibTeX
@misc{e2026approach,
title = {{A New Approach to Universal Measurability}},
author = {Reznichenko E. A. and Sadovnichiy Yu. V},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14414},
eprint = {2609.14414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14414},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A New Approach to Universal Measurability
AU - Reznichenko E. A.
AU - Sadovnichiy Yu. V
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14414
UR - https://arxiv.org/abs/2609.14414
AB - 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.
ER -