Invariant measures on locally compact spaces and a topological characterization of unimodular Lie groups. (English) Zbl 0668.22001

Let X be a locally compact space and let G be a group of homeomorphisms of X such that every G-orbit is dense in X. The author studies a necessary condition and a sufficient one for the existence of a nonzero G-invariant Borel measure on X. The main result is as follows. We denote by (B:A) the minimal number of G-images of the set A covering the set B, and by [B:A] the maximal number of pairwise disjoint G-images of A which are contained in B. Then the existence of \(\alpha\in (0,1)\) such that the inequalities \(\alpha\) (B:A)\(\leq [B:A]\leq (B:A)\) are satisfied for sufficiently many Borel sets guarantees the existence and uniqueness of a nonzero G-invariant measure. The results are applied to give a topological characterization of unimodular Lie groups.
Reviewer: H.Fujiwara


22E15 General properties and structure of real Lie groups
43A05 Measures on groups and semigroups, etc.
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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