PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 86, Number 3, November 1982 SEMIREGULARINVARIANT MEASURES ON ABELIANGROUPS ANDRZEJ PELC 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. We consider countably additive, nonnegative extended real-valued measures which are not identically equal to zero. Such a measure is called universal on a set X if it is defined on all subsets of X and it is called semiregular if every set of positive measure contains a set of positive finite measure. A universal measure m on a group (G, ° ) is invariant if m(a ° A) — m(A) for every a G G, A CG. A measure is called «-additive if the union of less than k sets of measure 0 has measure 0. A cardinal « is called real-valued measurable if there exists a finite universal «-additive measure on « which vanishes on singletons.1 It is well known that the existence of a real-valued measurable cardinal is unprovable in usual set theory with choice. Erdös and Mauldin [2] proved that if (G, ° ) is an uncountable group then there is no a-finite invariant universal measure on (G, ° ). Kannan and Raju [3] asked whether every invariant semiregular universal measure on a group is necessarily a multiple of the counting measure. It is clear that any measure providing a counter- example has to vanish on singletons. We give the following solution to the above problem. Theorem 1. The following are equivalent: (*) Every universal invariant semiregular measure on a group is a multiple of the counting measure. (**) There does not exist a real valued measurable cardinal. Proof.2 Assume that (*) does not hold. Hence there exists a universal invariant semiregular measure on a group, vanishing on singletons. Let A be any set of Received by the editors July 1, 1981. 1980 Mathematics Subject Classification. Primary 43A05; Secondary 28C10, 03E55. Key words and phrases. Universal invariant measure, group, real-valued measurable cardinal. 'According to the common habit in modern set theory we identify every ordinal with the set of its predecessors and cardinals with initial ordinals. 2The present version of the proof is simpler than the original one thanks to the referee's suggestion. ©1982 American Mathematical Society 0002-9939/82/0000-0325/S02.00 423 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use