The existence of nonmeasurable sets for invariant measures

Penconek, Marcin; Zakrzewski, Piotr

doi:10.1090/S0002-9939-1994-1182704-6

The existence of nonmeasurable sets for invariant measures
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by Marcin Penconek and Piotr Zakrzewski

Proc. Amer. Math. Soc. 121 (1994), 579-584

DOI: https://doi.org/10.1090/S0002-9939-1994-1182704-6

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Abstract:

We prove that if G is a locally compact Polish group acting in a reasonable way on a set X, then for every countably additive, $\sigma$-finite, G-invariant measure on X there exist nonmeasurable sets. In particular, the latter is true when X is a compact, metric, metrically homogeneous space, and G is the group of its isometries.

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The uniqueness of Haar measure and set theory