# Measure Theory and Integration

• Horst Osswald
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 510)

## Abstract

Loeb measures have been applied in various fields of real analysis. In his fundamental paper  Peter Loeb has given the first applications to probability theory. Also developed at that time (and published later in ) 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  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 kT. 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
$$A\left( {\omega ,t} \right): = \sum\limits_{s < t} {\omega \left( s \right)} \frac{1} {{\sqrt H }};$$
At) 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 tT with t ω 1, then $$A\left( {\omega ,t} \right) = \left( {t - 2} \right)\frac{1} {{\sqrt H }}$$ for each tT. Therefore At) may be unlimited.

## Keywords

Brownian Motion Measure Theory Standard Part Malliavin Calculus Internal Filtration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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