Measure reducibility of countable Borel equivalence relations. (English) Zbl 1390.54036

The main subject of investigation in this paper is (Borel-)reducibility of (Borel-)equivalence relations. If \(E\) and \(F\) are equivalence relations on \(X\) and \(Y\) respectively then \(\phi:X\to Y\) is a reduction of \(E\) to \(F\) if \(x\mathrel{E}y\) is equivalent to \(\phi(x)\mathrel{F}\phi(y)\) whenever \(x,y\in X\). The adverb ‘measure’ in ‘measure reducibility’ means that for every Borel probability measure on \(X\) there is a Borel set, \(B\), of full measure such that the restriction of \(E\) to \(B\) is Borel reducible to \(F\).
The main results of the paper are about the structure of non-measure-hyperfinite, projectively separable, treeable, and countable Borel equivalence relations, both individual relations and the class as a whole. The adjectives constitute quite a mouthful and rather than repeating all definitions here I refer to the authors’ excellent seven-page introduction to this paper for more information.
Reviewer: K. P. Hart (Delft)


54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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