Topological measure theory, with applications to probability
David Pollard · Bulletin of the Australian Mathematical Society · 1977
The work carried out for this thesis was motivated by a belief that the methods of topological measure theory could be more widely applied in the theory of probability. As my introduction to the subject was through the field of weak convergence, much of the thesis has developed out of the study of problems in that area; but this has often also involved consideration of the topological, measure theoretic and functional analytic background material. For example, the integral representation theory of Chapter 2 grew out of the study of relative compactness in the topology of weak convergence; the latter is the subject of Chapter 3. But the results of Chapter 2 are also of interest in their own right as they form the basis of a unified approach to Riesz type integral representation theorems. While these investigations were being carried out it became apparent that one particular concept has a most important part to play in topological measure theory. This so-called T-additivity property is examined in Chapter 4. It constitutes an intermediate stage in the progression from countable additivity to the stronger Radon measure concept. It often seems to be the minimal condition for compatibility between the topological and measure theoretic structures; this view is supported by the results of Chapter 4. The last two chapters contain some of the other applications of this research to probability theory. In Chapter 5 problems related to the existence and weak convergence of random measures on locally compact spaces are considered; and in Chapter 6 some aspects of the theory of Markov chains on topological state spaces are discussed. Weak convergence arguments are prominent in both chapters.