Congruence-invariant measures in uniform spaces

Abstract

In the following paper, we shall define a congruence relation among subsets of a given locally compact uniform space X, and then demonstrate a method for constructing a well-behaved congruence-invariant measure on X. In particular: if is compact, then the measure of will be 1. Another special case will be of interest: if every two points of have congruent neighborhoods, then the measure will be nontrivial and its support will be X. Thus, the method will yield Haar measure on a locally compact group. In these respects, our construction is superior to that of Appert [1]: his measure must be zero on any countable space. Although his procedures can be adjusted to obviate that defect, they do not seem to lead to general proofs of nontriviality in the cases mentioned above. We shall employ the axiom of choice, and we shall not concern ourselves with uniqueness theorems. The methods of Banach [2] and Loomis [3] are more satisfactory in these respects. In compensation, however, we shall be able to demonstrate the existence and nontriviality of our measure for spaces which do not satisfy their conditions. Such is the case not only for most compact spaces, but also for a significant class of our homogeneous spaces. (This will be demonstrated in ?6.) Our constructions generalize easily to any uniform space, but the results seem to have little content unless the space is at least locally totally-bounded. In the interests of stripping down the arguments, I shall restrict my attention somewhat further, and consider only locally compact spaces. In the sequel, then, X will denote a fixed locally compact uniform space. Its uniformities (entourages) will be denoted by u, v, etc. We shall use the following terminology: