Extension of kappa-additive measures
Abstract
We establish an extension theorem for kappa-additive probability measures, where kappa is an arbitrary uncountable cardinal. Starting from a <kappa-algebra, we define an outer measure using coverings of cardinality at most kappa and characterise measurable sets by approximation in outer measure of symmetric differences. We prove that these sets form a kappa-algebra carrying a complete kappa-additive extension of the original measure. The extension is unique on the generated kappa-algebra, and its completion is the minimal domain admitting a complete kappa-additive extension. We also establish attainment of the covering infimum and show why the closure assumption on the initial algebra is essential. Examples include a strict extension of classical Lebesgue measure under the additional hypothesis add N = aleph_1 < cov N.
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BibTeX
@misc{dzhenzher2026extension,
title = {{Extension of kappa-additive measures}},
author = {Sviatoslav V. Dzhenzher},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02729},
eprint = {2611.02729},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02729},
}
RIS
TY - UNPB
TI - Extension of kappa-additive measures
AU - Sviatoslav V. Dzhenzher
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02729
UR - https://arxiv.org/abs/2611.02729
AB - We establish an extension theorem for kappa-additive probability measures, where kappa is an arbitrary uncountable cardinal. Starting from a <kappa-algebra, we define an outer measure using coverings of cardinality at most kappa and characterise measurable sets by approximation in outer measure of symmetric differences. We prove that these sets form a kappa-algebra carrying a complete kappa-additive extension of the original measure. The extension is unique on the generated kappa-algebra, and its completion is the minimal domain admitting a complete kappa-additive extension. We also establish attainment of the covering infimum and show why the closure assumption on the initial algebra is essential. Examples include a strict extension of classical Lebesgue measure under the additional hypothesis add N = aleph_1 < cov N.
ER -