The existence of universal invariant semiregular measures on groups
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- by Piotr Zakrzewski PDF
- Proc. Amer. Math. Soc. 99 (1987), 507-508 Request permission
Abstract:
A nonnegative, countably additive, extended real-valued measure is universal on a set $X$ iff it is defined on all subsets of $X$, and is semiregular iff every set of positive measure contains a subset of positive finite measure. We prove that on every group of sufficiently large cardinality there exists a universal invariant semiregular measure vanishing on singletons. Thus we give complete solutions to the problems stated by Kannan and Raju [4] and Pelc [5].References
-
F. Drake, Set theory. An introduction to large cardinals, North-Holland, Amsterdan, 1974.
- A. Hulanicki, Invariant extensions of the Lebesgue measure, Fund. Math. 51 (1962/63), 111–115. MR 142709, DOI 10.4064/fm-51-2-111-115
- 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
- V. Kannan and S. Radhakrishneswara Raju, The nonexistence of invariant universal measures of semigroups, Proc. Amer. Math. Soc. 78 (1980), no. 4, 482–484. MR 556617, DOI 10.1090/S0002-9939-1980-0556617-4
- Andrzej Pelc, Semiregular invariant measures on abelian groups, Proc. Amer. Math. Soc. 86 (1982), no. 3, 423–426. MR 671208, DOI 10.1090/S0002-9939-1982-0671208-4
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 507-508
- MSC: Primary 43A05; Secondary 03E55, 28C10
- DOI: https://doi.org/10.1090/S0002-9939-1987-0875389-3
- MathSciNet review: 875389