On sets nonmeasurable with respect to invariant measures

Solecki, Sławomir

doi:10.1090/S0002-9939-1993-1159177-1

On sets nonmeasurable with respect to invariant measures
HTML articles powered by AMS MathViewer

by Sławomir Solecki PDF

Proc. Amer. Math. Soc. 119 (1993), 115-124 Request permission

Abstract:

A group $G$ acts on a set $X$, and $\mu$ is a $G$-invariant measure on $X$. Under certain assumptions on the action of $G$ and on $\mu$ (e.g., $G$ acts freely and is uncountable, and $\mu$ is $\sigma$-finite), we prove that each set of positive $\mu$-measure contains a subset nonmeasurable with respect to any invariant extensions of $\mu$. We study the case of ergodic measures in greater detail.

References

Albert Ascherl and Jürgen Lehn, Two principles for extending probability measures, Manuscripta Math. 21 (1977), no. 1, 43–50. MR 453951, DOI 10.1007/BF01176900
Dietrich Bierlein, Über die Fortsetzung von Wahrscheinlichkeitsfeldern, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/63), 28–46 (German). MR 149533, DOI 10.1007/BF00531770
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
Paul Erdős and R. Daniel Mauldin, The nonexistence of certain invariant measures, Proc. Amer. Math. Soc. 59 (1976), no. 2, 321–322. MR 412390, DOI 10.1090/S0002-9939-1976-0412390-0
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
A. B. Harazišvili, Certain types of invariant measures, Dokl. Akad. Nauk SSSR 222 (1975), no. 3, 538–540 (Russian). MR 0382603
Andrzej Pelc, Invariant measures and ideals on discrete groups, Dissertationes Math. (Rozprawy Mat.) 255 (1986), 47. MR 872392
C. Ryll-Nardzewski and R. Telgársky, The nonexistence of universal invariant measures, Proc. Amer. Math. Soc. 69 (1978), no. 2, 240–242. MR 466494, DOI 10.1090/S0002-9939-1978-0466494-9
Piotr Zakrzewski, The existence of universal invariant semiregular measures on groups, Proc. Amer. Math. Soc. 99 (1987), no. 3, 507–508. MR 875389, DOI 10.1090/S0002-9939-1987-0875389-3