A generalized topological measure theory

Abstract

The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra C b {C^b} of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra A A of bounded real-valued functions. Necessary and sufficient conditions are given for A ∗ {A^ \ast } to be represented in the natural way by a space of regular finitely-additive set functions. The concepts of additivity and tightness for these set functions are considered and some remarks about weak convergence are made.