Chapter 39 Non-standard analysis with applications to economics

Abstract

This chapter presents an introduction to nonstandard analysis and surveys its applications in mathematical economics. Nonstandard analysis is a mathematical technique, which has been widely used in diverse areas in pure and applied mathematics, including probability theory, mathematical physics, and functional analysis. It is used to formalize most areas of modern mathematics, including real and complex analysis, measure theory, probability theory, functional analysis, and point set topology; algebra is less amenable to nonstandard treatments, but even there significant applications have been found. The primary goal is to provide a careful development of nonstandard methodology in sufficient detail to allow using it in diverse areas in mathematical economics. This requires a careful study of the nonstandard treatment of real analysis, measure theory, and topological spaces. To accommodate this extended treatment of methodology the survey of work to date using nonstandard methods in mathematical economics is briefly reviewed in the chapter.