Loeb measure

Summary

A Loeb measure is a genuine countably additive measure built from a finitely additive measure inside a nonstandard model of analysis. One takes the standard part of the nonstandard measure and extends it to a σ-algebra. Introduced by Peter Loeb, it lets hyperfinite objects, such as a finite set with an infinite number of points, serve as ordinary measure spaces. It is the main tool of nonstandard measure theory and gives nonstandard constructions of Lebesgue measure and of Brownian motion.