Semiregular invariant measures on abelian groups
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- by Andrzej Pelc
- Proc. Amer. Math. Soc. 86 (1982), 423-426
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671208-4
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Abstract:
A nonnegative countably additive, extended real-valued measure is called semiregular if every set of positive measure contains a set of positive finite measure. V. Kannan and S. R. Raju [3] stated the problem of whether every invariant semiregular measure defined on all subsets of a group is necessarily a multiple of the counting measure. We prove that the negative answer is equivalent to the existence of a real-valued measurable cardinal. It is shown, moreover, that a counterexample can be found on every abelian group of real-valued measurable cardinality.References
- F. Drake, Set theory. An introduction to large cardinals, North-Holland, Amsterdam, 1974.
- 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
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 423-426
- MSC: Primary 03E35; Secondary 03E55, 28C10
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671208-4
- MathSciNet review: 671208