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Measurability properties of sets of Vitali’s type. (English) Zbl 0795.28010
Let $$G$$ 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.
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.
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