# Measure Theory and Integration

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## 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*(ω*t*) 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*(ω*t*) may be unlimited.## Keywords

Brownian Motion Measure Theory Standard Part Malliavin Calculus Internal Filtration
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