×

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 \(\mu\).

MSC:

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
PDFBibTeX XMLCite