Measurability properties of sets of Vitali’s type.

*(English)*Zbl 0795.28010Let \(G\) be a group acting on a fixed set \(X\) and \(\mu\) a \(G\)-invariant countably additive measure on \(X\). If \(H\) is a subgroup of \(G\), then \(H\)- selector means a set having exactly one point in common with each orbit of \(H\). The action of \(G\) is \(\mu\)-free if \(\mu^*(\{x\in X: hx= x\})= 0\) for any \(h\in G\backslash \{e\}\) (\(e=\) the identity of \(G\)). The cardinality of a set \(A\) is denoted by \(| A|\). Also for a cardinal number \(\lambda\), \(cf (\lambda)=\min\{K: K\) is an ordinal and \(\exists f: K\to \lambda\), \(\lambda=\bigcup_{\alpha< K} f(\alpha)\}\). The following two theorems have been proved.

Theorem 1. Let \(G\) be uncountable and let \(\mu\) be \(\sigma\)-finite. Suppose \(G\) acts \(\mu\)-freely on \(X\). Then there exists a countable subgroup \(H\) of \(G\) such that each \(H\)-selector is nonmeasurable with respect to any invariant extension of \(\mu\). Theorem 2. Assume \(cf(| G|)> \omega\). Suppose also that \(G\) acts freely on \(X\). Let \(\mu\) be \(\sigma\)-finite and ergodic. Then there exists an invariant extension \(\bar\mu\) of \(\mu\) such that for each subgroup \(H\) of \(G\) with \(| H|= | G|\) there is a \(\bar \mu\)-measurable \(H\)-selector.

Theorem 1. Let \(G\) be uncountable and let \(\mu\) be \(\sigma\)-finite. Suppose \(G\) acts \(\mu\)-freely on \(X\). Then there exists a countable subgroup \(H\) of \(G\) such that each \(H\)-selector is nonmeasurable with respect to any invariant extension of \(\mu\). Theorem 2. Assume \(cf(| G|)> \omega\). Suppose also that \(G\) acts freely on \(X\). Let \(\mu\) be \(\sigma\)-finite and ergodic. Then there exists an invariant extension \(\bar\mu\) of \(\mu\) such that for each subgroup \(H\) of \(G\) with \(| H|= | G|\) there is a \(\bar \mu\)-measurable \(H\)-selector.

Reviewer: K.C.Ray (Kalyani)

##### MSC:

28C10 | Set functions and measures on topological groups or semigroups, Haar measures, invariant measures |

43A05 | Measures on groups and semigroups, etc. |