Measure Theory and Integration

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


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 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.


Brownian Motion Measure Theory Standard Part Malliavin Calculus Internal Filtration 
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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Horst Osswald
    • 1
  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany

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