TABLE OF CONTENTS We first carry out Andersen s construction of Brownian motion. This is done by a very intuitive approach: a hyper finite random walk. We verify that the standard part of this random walk satisfies all required properties. This is proved by purely elementary combinatorial arguments.
We also show variants of Anderson s model. The first one is a *continuous model obtained from an hyperfinite product of copies of a Gaussian measure with infinitesimal variance. The other is obtained from the uniform measure on an infinite dimensional unit sphere.
Historically, the first rigorous mathematical construction of Brownian motion was due to Wiener [Wiener, 1923], although Bachelier used a naive version of Brownian motion in [Bachelier, 1900]. See [Karatzas and Shreve, 1991] for a modern standard treatment of the subject.
We only consider Anderson s construction of one-dimensional Brownian motion. The n-dimensional Brownian motion can be obtained by taking the product of n independent copies. We continue to use the hyperfinite time line defined in Definition 7.3.1, i.e.
where ? t=1 /N for some 
. (Again, we are free to assume that N is a factorial for the convenience that 
includes all rational points from the unit interval.) models the unit interval [0,1], which is thought of as the time line for stochastic processes.
We let 
, A=* P( ?), the algebra of internal subsets of ?, and ? the counting...
