Chapter 1 - Mathematical Preliminaries

Abstract

This chapter presents some mathematical preliminaries. It is a mere collection of definitions and theorems given without a proof. The chapter discusses the relevant material in three sections: analysis and topology, probability theory, and algebra. The first section covers such basic topics as linear spaces, normed and metric spaces, point-set topology, measures, and the different notions of convergence encountered in analysis. The second section covers the probability measures, distribution functions, random variables, and their interconnections. It defines the Lebesgue spaces, states the Riesz Representation Theorem, and gives a brief overview of Markov processes and Markov chains. The last section deals with diagrams, semigroups, groups, and semigroup and group endomorphisms and introduces free semigroups and free groups. It also provides a brief review of category theory and direct and inverse limits.