Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f: S → ℝ is of the form f(x,y) = u(x) + v(y), where u: X → ℝ and v: Y → ℝ are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-1-6,
author = {Benjamin D. Miller},
title = {Coordinatewise decomposition, Borel cohomology, and invariant measures},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {81-94},
zbl = {1097.03042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-1-6}
}
Benjamin D. Miller. Coordinatewise decomposition, Borel cohomology, and invariant measures. Fundamenta Mathematicae, Tome 189 (2006) pp. 81-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-1-6/