Coordinatewise decomposition, Borel cohomology, and invariant measures

Benjamin D. Miller¹

¹ Department of Mathematics University of California 520 Portola Plaza Los Angeles, CA, 90095-1555, U.S.A.

Fundamenta Mathematicae, Tome 191 (2006) no. 1, pp. 81-94.

Résumé

Given Polish spaces $X$ and $Y$ and a Borel set $S \subseteq X \times Y$ with countable sections, we describe the circumstances under which a Borel function $f : S \rightarrow \mathbb R$ is of the form $f(x,y) = u(x) + v(y)$, where $u : X \rightarrow \mathbb R$ and $v : Y \rightarrow \mathbb R$ 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 $\sigma$-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.

Détail

DOI : 10.4064/fm191-1-6

Keywords: given polish spaces and borel set subseteq times countable sections describe circumstances under which borel function rightarrow mathbb form where rightarrow mathbb rightarrow mathbb borel turns out special problem determining whether real valued borel cocycle countable borel equivalence relation coboundary several glimm effros style dichotomies solution problem terms certain sigma finite measures underlying space main technical ingredient characterization existence type iii measures given cocycle