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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