1 - Metric measure spaces

Summary

Abstract

The contents of this chapter are introductory and covered in many standard books on probability theory, but perhaps not all conveniently in one place. In Section 1.1 we give a summary of results concerning probability measures on compact metric spaces. Section 1.2 concerns the existence of invariant measure on a compact metric group, which we later use to construct random matrix ensembles. In Section 1.3, we resume the general theory with a discussion of weak convergence of probability measures on (noncompact) Polish spaces; the results here are technical and may be omitted on a first reading. Section 1.4 contains the Brunn–Minkowski inequality, which is our main technical tool for proving isoperimetric and concentration inequalities in subsequent chapters. The fundamental example of Gaussian measure and the Gaussian orthogonal ensemble appear in Section 1.5, then in Section 1.6 Gaussian measure is realised as the limit of surface area measure on the spheres of high dimension. In Section 1.7 we state results from the general theory of metric measure spaces. Some of the proofs are deferred until later chapters, where they emerge as important special cases of general results. A recurrent theme of the chapter is weak convergence, as defined in Sections 1.1 and 1.3, and which is used throughout the book. Section 1.8 shows how weak convergence gives convergence for characteristic functions, cumulative distribution functions and Cauchy transforms.