Vol. 44, No. 1, 1973

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ISSN: 0030-8730
Invariant measures and the converse of Haar’s theorem on semitopological semigroups

Arunava Mukherjea and Nicolas A. Tserpes

Vol. 44 (1973), No. 1, 251–262

According to the converse of Haar’s theorem, a complete separable metric group which admits a locally finite nonzero r-invariant measure is locally compact. It is proved in this paper that a complete separable metric semitopological (resp. topological) semigroup which admits a finite (resp. possibly infinite) nonzero r- and l-invariant measure is a compact (resp. locally compact) topological group.

The structure of idempotent probability measures and finite r-invariant measures has also been given in the case of locally compact semitopological semigroups. Also it is shown in this paper that for a wide class of nonabelian locally compact semitopological semigroups, admissibility of a twoside invariant measure is equivalent to embeddability in a group.

Mathematical Subject Classification 2000
Primary: 22A15
Secondary: 43A05
Received: 24 August 1971
Revised: 28 June 1972
Published: 1 January 1973
Arunava Mukherjea
Nicolas A. Tserpes