Chapter 32 - Nonstandard Analysis and Measure Theory

Abstract

This chapter examines nonstandard analysis and measure theory. Mathematicians have used infinitesimal numbers for thousands of years as a means of expressing intuitive mathematical ideas. Brownian motion is the random motion of microscopic particles suspended in a liquid or gas. An important part of mathematical probability is the construction of mathematical models for this motion. An intuitive model for Brownian motion works with a particle that performs a random walk with steps of infinitesimal length. For this model, one divides time into infinitesimal intervals and in each time interval, the particle moves in a straight line over an infinitesimal distance. An object in the superstructure is called internal, if it is an element of the nonstandard extension of a standard object. The notion of a hyperfinite subset is applicable for the extension of any standard set. Hyperfinite sets are extremely useful because combinatorial results continue to be valid even when the external cardinality is infinite. A lattice approach to develop measure theory is also elaborated in the chapter.