Introduction

This text provides a concise introduction, suitable for a one-semester special topics

course, to the remarkable properties of Gaussian measures on both finite and infinite

dimensional spaces. It begins with a brief resumé of probabilistic results in which Fourier

analysis plays an essential role, and those results are then applied to derive a few basic

facts about Gaussian measures on finite dimensional spaces. In anticipation of the analysis

of Gaussian measures on infinite dimensional spaces, particular attention is given to those

properties of Gaussian measures that are dimension independent, and Gaussian processes

are constructed. The rest of the book is devoted to the study of Gaussian measures on

Banach spaces. The perspective adopted is the one introduced by I. Segal and developed

by L. Gross in which the Hilbert structure underlying the measure is emphasized.

Target Audience and Applications

The contents of this book should be accessible to either undergraduate or graduate

students who are interested in probability theory and have a solid background in Lebesgue

integration theory and a familiarity with basic functional analysis. Although the focus is

on Gaussian measures, the book introduces its readers to techniques and ideas that have

applications in other contexts.