A generalized topological measure theory

Kirk, R. B.; Crenshaw, J. A.

doi:10.1090/S0002-9947-1975-0369648-7

A generalized topological measure theory
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by R. B. Kirk and J. A. Crenshaw

Trans. Amer. Math. Soc. 207 (1975), 189-217

DOI: https://doi.org/10.1090/S0002-9947-1975-0369648-7

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Abstract:

The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra ${C^b}$ of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra $A$ of bounded real-valued functions. Necessary and sufficient conditions are given for ${A^ \ast }$ 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.

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