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Abstract

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.