Measure Theory and Integration

Abstract

Loeb measures have been applied in various fields of real analysis. In his fundamental paper [9] Peter Loeb has given the first applications to probability theory. Also developed at that time (and published later in [10]) was an application constructing representing measures in potential theory. (See Section 3.12.2.) The next convincing example of the usefulness of Loeb measures is Bob Anderson’s [2] construction of Brownian motion from a hyperfinite model of tossing an unbiased coin. Let us briefly sketch Anderson’s approach: Fix an unlimited positive integer H} and put T := { 1,...,H} This set T is infinite, but * finite, and can be interpreted as a “time line”, which is closely related to the continuous time line [0,1], because each real number between 0 and 1 is infinitely close to some $\frac{k} {H}$ with k ∈T. Let { -1, l}^T be the set of all internal H-tuples of the numbers -1 and 1. As noted in Section 3.12, this set can be interpreted as the set of all outcomes of tossing a coin H-times. Anderson defines an internal process A : -1,1 _T × T → *ℝ by setting

$$ A\left( {\omega ,t} \right): = \sum\limits_{s < t} {\omega \left( s \right)} \frac{1} {{\sqrt H }}; $$

A (ωt) can be understood as the profit (or loss) at time t during the game ω ∈ if the gamblers are playing for the infinitesimal stake of $\frac{1} {{\sqrt H }}$ (dollar, mark, euro, lira pound sterling, it always remains an infinitestimal amount of money). for example, if ω (1) = -1 and ω (t)= 1 for each t ∈ T with t ω 1, then $A\left( {\omega ,t} \right) = \left( {t - 2} \right)\frac{1} {{\sqrt H }}$ for each t ∈ T. Therefore A (ωt) may be unlimited.