PreprintCC BY 4.0arXiv
A note on Rainwater's Theorem
Abstract
We extend Rainwater's theorem that, for Baire measures on a compact Hausdorff space,a measure singular with respect to any measure from a w*-compact convex set M of probability measures sits on a Baire set which is μ-null for every μ∈ M. Our main result shows that this statement holds under the weaker assumption of convex analyticity of M. In the standard way, this extends the Glicksberg--König--Seever decomposition theorem. We also give conditions for our results to hold for sets of signed measures.
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BibTeX
@misc{preiss2026note,
title = {{A note on Rainwater's Theorem}},
author = {David Preiss},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36306},
eprint = {2609.36306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36306},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A note on Rainwater's Theorem
AU - David Preiss
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36306
UR - https://arxiv.org/abs/2609.36306
AB - We extend Rainwater's theorem that, for Baire measures on a compact Hausdorff space,a measure singular with respect to any measure from a w^*-compact convex set M of probability measures sits on a Baire set which is μ-null for every μ∈ M. Our main result shows that this statement holds under the weaker assumption of convex analyticity of M. In the standard way, this extends the Glicksberg--König--Seever decomposition theorem. We also give conditions for our results to hold for sets of signed measures.
ER -