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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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References:
 [1] Albert Ascherl and Jürgen Lehn, Two principles for extending probability measures, Manuscripta Math. 21 (1977), no. 1, 43 – 50. · Zbl 0368.60005 · doi:10.1007/BF01176900 · doi.org [2] Dietrich Bierlein, Über die Fortsetzung von Wahrscheinlichkeitsfeldern, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), 28 – 46 (German). , https://doi.org/10.1007/BF00531770 Dietrich Bierlein, Die Konstruktion eines Masses \?\mid\?(\?\times \?) zu vorgegebenem Marginalmass \?\mid\? mit \?(\?$$_{0}$$)=1 für eine vorgegebene Menge \?$$_{0}$$, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), 126 – 140 (German). · Zbl 0117.12804 · doi:10.1007/BF01844415 · doi.org [3] Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. Ryszard Engelking, General topology, PWN — Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. [4] Paul Erdős and R. Daniel Mauldin, The nonexistence of certain invariant measures, Proc. Amer. Math. Soc. 59 (1976), no. 2, 321 – 322. · Zbl 0361.28013 [5] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802 [6] A. B. Harazišvili, Certain types of invariant measures, Dokl. Akad. Nauk SSSR 222 (1975), no. 3, 538 – 540 (Russian). [7] Andrzej Pelc, Invariant measures and ideals on discrete groups, Dissertationes Math. (Rozprawy Mat.) 255 (1986), 47. · Zbl 0625.04009 [8] C. Ryll-Nardzewski and R. Telgársky, The nonexistence of universal invariant measures, Proc. Amer. Math. Soc. 69 (1978), no. 2, 240 – 242. · Zbl 0352.28010 [9] Piotr Zakrzewski, The existence of universal invariant semiregular measures on groups, Proc. Amer. Math. Soc. 99 (1987), no. 3, 507 – 508. · Zbl 0616.43001
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