Abstract
The aim of this article is to explain how and why the Loeb measure construction can be applied to problems which arise outside of nonstandard analysis. The Loeb measure and the necessary background from nonstandard analysis are presented in the paper of Lindstrøm in this volume. Most of the applications of Loeb measures in probability theory fit the following pattern. First, lift the original classical problem to a hyperfinite setting. Second, make some hyperfinite computations. Third, take standard parts of everything in sight to obtain the desired classical result. In keeping with this pattern, we shall concentrate on hyperfinite Loeb spaces, that is, Loeb spaces on hyperfinite sets. As we go along, we shall present several examples of applications of hyperfinite Loeb spaces to probability theory. However, our main emphasis will be on the general theory which underlies the applications.
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