ABSTRACT
Loeb measures, discovered by Peter Loeb in 1975, are very rich yet tractable measure spaces, which play a central and powerful role in many applications of nonstandard analysis – in measure and probability theory, stochastic analysis, functional analysis, mathematical physics, economics and mathematical finance theory. This chapter provides an introduction to the basics of Loeb measure and integration theory, designed to make the extensive literature that uses the notions accessible. Loeb integration theory is simply the theory of the integral in the classical sense with respect to Loeb measures. A Loeb measure is a measure constructed from a nonstandard measure by the following construction of Loeb. Loeb measures have been used extensively for representation of measures and for construction of measures for a wide variety of purposes. The hyperfinite difference approach has been used to great affect in the solution of It o stochastic differential equations, based on Anderson’s hyperfinite random walk construction of Brownian motion and the Ito integral.
