Preface

There are a myriad of books about probability theory already on the mar-

ket. Nonetheless, a few years ago Sergei Gelfand asked if I would write a

probability theory book for advanced undergraduate and beginning gradu-

ate students who are interested in mathematics. He had in mind an updated

version of the first volume of William Feller’s renowned An Introduction to

Probability Theory and Its Applications [3]. Had I been capable of doing

so, I would have loved to oblige, but, unfortunately, I am neither the math-

ematician that Feller was nor have I his vast reservoir of experience with

applications. Thus, shortly after I started the project, I realized that I

would not be able to produce the book that Gelfand wanted. In addition,

I learned that there already exists a superb replacement for Feller’s book.

Namely, for those who enjoy combinatorics and want to see how probability

theory can be used to obtain combinatorial results, it is hard to imagine

a better book than N. Alon and J. H. Spencer’s The Probabilistic Method

[1]. For these reasons, I have written instead a book that is a much more

conventional introduction to the ideas and techniques of modern probabil-

ity theory. I have already authored such a book, Probability Theory, An

Analytic View [9], but that book makes demands on the reader that this

one does not. In particular, that book assumes a solid grounding in analy-

sis, especially Lebesgue’s theory of integration. In the hope that it will be

appropriate for students who lack that background, I have made this one

much more self-contained and developed the measure theory that it uses.

Chapter 1 contains my attempt to explain the basic concepts in proba-

bility theory unencumbered by measure-theoretic technicalities. After intro-

ducing the terminology, I devote the rest of the chapter to probability theory

on finite and countable sample spaces. In large part because I am such a

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