On selectors nonmeasurable with respect to quasiinvariant measures. (English) Zbl 0879.28028

Let \(E\) be a nonempty set and \(G\) a group of transformations of \(E\). If \(S\) is a \(\sigma\)-algebra of subsets of \(E\) and \(\mu\) a measure defined on \(S\), then \(\mu\) is called a \(G\)-quasiinvariant measure if
(a) the \(\sigma\)-algebra \(S\) is a \(G\)-invariant class of sets;
(b) if \(X\in S\), \(\mu (X)=0\), then \(\mu (g(X))=0\) for each \(g\in G\).
Let \(H\) be a subgroup of \(G\) and let \(\mathfrak {M}_H\) be the partition of \(E\) consisting of all \(H\)-orbits. A subset \(Y\subset E\) is called \(H\)-selector if \(Y\cap M\) contains exactly one element for each \(M\in \mathfrak {M}_H\). A group \(G\) is said to act freely in the space \(E\) with respect to the given measure \(\mu\) if for any two distinct transformations \(g\) and \(h\) the equality \[ \mu^\ast (\{x\in E:\;g(x)=h(x)\})=0 \] holds, where \(\mu^\ast\) denotes the outer measure associated with \(\mu\).
The author proves the following theorem: Let \(E\) be a set and \(G\) be an uncountable group of transformations of \(E\). Let \(\mu\) be a nonzero \(\sigma\)-finite \(G\)-quasiinvariant measure defined on some \(\sigma\)-algebra of subsets of \(E\). Suppose that \(G\) acts freely in \(E\) with respect to \(\mu\). Let \(H\) be a countable subgroup of \(G\) and denote by \(\mathfrak {M}_H\) the partition of \(E\) consisting of all \(H\)-orbits. Then there exists a subfamily of \(\mathfrak {M}_H\) such that all its selectors are nonmeasurable with respect to \(\mu\).


28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets