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On sets nonmeasurable with respect to invariant measures. (English) Zbl 0784.28006
A. B. Harazisvili [Doklady Akad. Nauk SSSR 222, 538-540 (1975; Zbl 0328.28011)] and independently P. Erdős and R. D. Mauldin [Proc. Am. Math. Soc. 59, 321-322 (1976; Zbl 0361.28013)] proved that given a \(\sigma\)-finite invariant measure \(m\) on an uncountable group there exists a non-measurable set. The present author answers in the affirmative a question of A. Pelc [Diss. Math. 255 (1986; Zbl 0625.04009)] whether the stronger conclusion may be drawn, namely, that every set of positive measure must contain a subset non-measurable with respect to any invariant extension of the given measure \(m\).
The above result is proved in the context of a group \(G\) operating on a set \(X\) with the \(\sigma\)-finite invariant measure \(m\) defined on the set \(X\). The group \(G\) is assumed to act \(m\)-freely, i.e., the outer measure of \(\{x\in X:hx=x\}\) must be equal to zero for any element \(h\) in \(G\) except for the identity element. The author also extracts the same conclusion under two other conditions, namely: if \(L^ 1(m)\) has a dense subset of cardinality less than the cardinality of the group \(G\), or if the measure \(m\) is ergodic and non-atomic.

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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