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Proceedings of the American Mathematical Society

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Semiregular invariant measures on abelian groups

Author: Andrzej Pelc
Journal: Proc. Amer. Math. Soc. 86 (1982), 423-426
MSC: Primary 03E35; Secondary 03E55, 28C10
MathSciNet review: 671208
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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 [Enhancements On Off] (What's this?)

  • [1] F. Drake, Set theory. An introduction to large cardinals, North-Holland, Amsterdam, 1974.
  • [2] P. Erdös and R. D. Mauldin, The nonexistence of certain invariant measures, Proc. Amer. Math. Soc. 59 (1976), 321-322. MR 0412390 (54:516)
  • [3] V. Kannan and S. Radhakrishnesvara Raju, The nonexistence of invariant universal measures on semigroups, Proc. Amer. Math. Soc. 78 (1980), 482-484. MR 556617 (81b:28013)

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Keywords: Universal invariant measure, group, real-valued measurable cardinal
Article copyright: © Copyright 1982 American Mathematical Society

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