Vol. 41, No. 1, 1972

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A characterization of a class of uniform spaces that admit an invariant integral

Gerald L. Itzkowitz

Vol. 41 (1972), No. 1, 123–141

In this paper we consider a class of uniform spaces that we temporarily call equihomogeneous spaces. These spaces were first considered by Y. Mibu in a paper On Measures Invariant Under Given Homeomorphism Groups of A Uniform Space. The reason for considering equihomogeneous spaces is that one can easily show the existence of a Haar type integral on them just by using an obvious modification of a standard existence type proof for the Haar integral on locally compact topological groups. We show that these spaces coincide in the locally compact case with the class of uniformly locally compact spaces considered by I. E. Segal in his paper Invariant Measures on Locally Compact Spaces that appeared in 1949. Our main theorem is that a locally compact equihomogeneous space is a locally compact topological homogeneous space and hence is a quotient of locally compact topological groups. We are therefore able to use the theory of A. Weil to deduce existence and uniqueness of an invariant integral for these spaces. These results seem to explain why no examples of spaces, satisfying Segal’s or Mibu’s conditions, aside from topological groups and their quotients have been found to date.

Mathematical Subject Classification
Primary: 28A70
Received: 2 November 1970
Published: 1 April 1972
Gerald L. Itzkowitz