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.
