We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-4,
author = {Petr Holick\'y},
title = {Borel sets with $\sigma$-compact sections for nonseparable spaces},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {139-154},
zbl = {1202.54027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-4}
}
Petr Holický. Borel sets with σ-compact sections for nonseparable spaces. Fundamenta Mathematicae, Tome 201 (2008) pp. 139-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-4/