This graduate-level probability textbook was originally published by World Scientific Publishing Co. in 2000 (subsequent printings 2003, 2005, 2006), with a second edition published in 2006 (subsequent printings 2007, 2009, 2010, 2011, 2013). It may be ordered for U.S. $33 (cheap!) directly from the publisher, or from e.g. amazon.ca or amazon.com or amazon.co.uk or indigo.ca.

Below are some reviews and
the preface
and second-edition preface
and table of contents.
See also the errata in postscript / PDF
(or the *first edition* errata in postscript / PDF).

NEW:There is now a free, public on-line solutions manual to all even-numbered exercises, by M. Soltanifar, L. Li, and myself. [See Exercise 13.4.6 supplement.]

(See also Evans and Rosenthal's introductory-level probability and statistics book, and an unexpected spoof video.)

Rosenthal, Jeffrey S.(3-TRNT)

A first look at rigorous probability theory.

World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xiv+177 pp. $24.00. ISBN 981-02-4322-7

This book is an introduction to probability theory using measure theory. It provides mathematically complete proofs of all the essential introductory results of probability and measure theory.

The book is divided into fifteen sections and two appendices. The first six sections contain the essential core of measure-theoretic probability theory: sigma-algebras; construction of probability measures; random variables; expected values; inequalities and laws of large numbers; and distributions of random variables. The following two sections introduce dynamic aspects of probability models: stochastic processes are introduced using gambling games as the motivating example and discrete Markov chains are discussed in some detail. The following section complements the results with measure-theoretic flavor by discussing and proving results such as the dominated convergence theorem and Fubini's theorem. Sections 10 to 14 contain a collection of further topics including weak convergence, characteristic functions (together with a proof of the central limit theorem), decomposition of probability laws, conditional probability and expectation and martingales. The final section then provides an appetizer for further topics in the subject of stochastic processes and applications. It contains material on Markov chains on general state spaces, diffusions and stochastic integrals, and the Black-Scholes formula. The appendices provide mathematical background and a guide to further reading.

The book is certainly well placed to establish itself as a core reading in measure-theoretic probability. However, a more complete and advanced book, such as [P. Billingsley, Probability and measure, Third edition, Wiley, New York, 1995; MR 95k:60001], might be needed as a complementary source for graduate students in mathematics and statistics. Furthermore, although the text contains a variety of excellent exercises, students from economics, computer science, engineering, etc., might find the addition of more applied examples and exercises beneficial.

I found this little book delightful reading and a worthwhile addition to the existing literature.

Reviewed by Rüdiger Kiesel

The reader will get basic ideas on most fundamental topics in probability theory in a detailed (as far as the proofs are concerned), mathematically rigorous and very readable way. [...] The author presents a very good selection on a mere 219 pages. [...]

Chapter 15 presents a nice heuristic introduction to Markov chains with general state space, continuous time Markov processes, Brownian motion, diffusions, and stochastic integrals.

Reviewed by Dalibor Volny

March 15, 2002

Reviewer: from California

This is a marvelous primer on measure-theoretic probability. I came across it a couple of years after taking a course based on Chung's famous text ("A Course in Prob. Theory") and found it to be an excellent book for review and remediation--that is, it helped me get a better overview of the material I had already learned and it helped me learn topics such as, say, uniform integrability, that didn't sink in too well the first time around.

According to the preface, the author prepared most of the book as supplemental class notes for the benefit of his students in a course whose main text was, if I recall correctly, Billingsley's excellent "Probability and Measure". The students were so enthusiastic about the usefulness of Professor Rosenthal's supplemental info that they insisted he publish it, despite his objection that the book wasn't original enough to warrant entry into an already crowded field. Well, the students made the right call: Rosenthal's clear and concise text will, I think, help almost any student learn measure-theoretic probability more efficiently. I'd also recommend it to folks who need a concise review of measure-theoretic probability.

(5 stars) Best Probability book ever!

July 10, 2006

Reviewer: Thomas R. Fielden (Portland, OR USA)

As a graduate student in mathematics I appreciate the rigorous and no nonsense treatment of the subject. I'm am using this text to study for my Ph.D. qualifying exam in statistics. It's explaining statistics in a language I understand.

(5 stars) A gem.

July 17, 2007

Reviewer: M. Henri De Feraudy (France)

This is my bedside book at the present time. It's compact, written with immense respect for the reader and even covers some financial applications.

It's recalling the measure theory I learned as an undergraduate with the right style.

So much better than some of the "Probabilty from dummies" I have put away.

When I finish the book I hope to move on to some of the heavier books with a clear idea of where I am going.

(5 stars) A delightful read and a great introduction.

June 12, 2009

Reviewer: A customer

I took this book from the library during a course in measure-theoretic probability, and how lucky I was to come across it!

A very well structured book, the choice of material (for an introduction) is excellent. As the title suggests, the book is rather rigorous (most results are with proofs, which helps understand the theory better), and at the same time the author does a good job at motivating the introduction of the mathematical concepts required to understand (rigorous) probability.

The best part is that, for any mathematician, this book will also be a lot of fun to read!

I would like to sincerely congratulate the author for making something this really good.

This text is intended to answer that need. It provides an introduction to rigorous (i.e., mathematically precise) probability theory using measure theory. At the same time, I have tried to make it brief and to the point, and as accessible as possible. In particular, probabilistic language and perspective are used throughout, with necessary measure theory introduced only as needed.

I have tried to strike an appropriate balance between rigorously covering the subject, and avoiding unnecessary detail. The text provides mathematically complete proofs of all of the essential introductory results of probability theory and measure theory. However, more advanced and specialised areas are ignored entirely or only briefly hinted at. For example, the text includes a complete proof of the classical Central Limit Theorem, including the necessary Continuity Theorem for characteristic functions. However, the Lindeberg Central Limit Theorem and Martingale Central Limit Theorem are only briefly sketched and are not proved. Similarly, all necessary facts from measure theory are proved before they are used. However, more abstract and advanced measure theory results are not included. Furthermore, the measure theory is almost always discussed purely in terms of probability, as opposed to being treated as a separate subject which must be mastered before probability theory can be studied.

I hesitated to bring these notes to publication. There are many other books available which treat probability theory with measure theory, and some of them are excellent. For a partial list see Subsection B.3 on page 169. (Indeed, the book by Billingsley was the textbook from which I taught before I started writing these notes. While much has changed since then, the knowledgeable reader will still notice Billingsley's influence in the treatment of many topics herein. The Billingsley book remains one of the best sources for a complete, advanced, and technically precise treatment of probability theory with measure theory.) In terms of content, therefore, the current text adds very little indeed to what has already been written. It was only the reaction of certain students, who found the subject easier to learn from my notes than from longer, more advanced, and more all-inclusive books, that convinced me to go ahead and publish. The reader is urged to consult other books for further study and additional detail.

There are also many books available (see Subsection B.2) which treat probability theory at the undergraduate, less rigorous level, without the use of general measure theory. Such texts provide intuitive notions of probabilities, random variables, etc., but without mathematical precision. In this text it will generally be assumed, for purposes of intuition, that the student has at least a passing familiarity with probability theory at this level. Indeed, Section 1 of the text attempts to link such intuition with the mathematical precision to come. However, mathematically speaking we will not require many results from undergraduate-level probability theory.

**Structure.**
The first six sections of this book could be
considered to form a "core" of essential material. After learning
them, the student will have a precise mathematical
understanding of probabilities and sigma-algebras; random variables,
distributions, and expected values; and inequalities and laws of large numbers.
Sections 7 and 8 then diverge into the theory of gambling games
and Markov chain theory.
Section 9 provides a
bridge to the more advanced topics of Sections 10 through 14, including
weak convergence, characteristic functions, the Central Limit Theorem,
Lebesgue Decomposition, conditioning, and martingales.

The final section, Section 15, provides a wide-ranging and somewhat less rigorous introduction to the subject of general stochastic processes. It leads up to diffusions, Ito's Lemma, and finally a brief look at the famous Black-Sholes equation from mathematical finance. It is hoped that this final section will inspire readers to learn more about various aspects of stochastic processes.

Appendix A contains basic facts from elementary mathematics. This appendix can be used for review and to gauge the book's level. In addition, the text makes frequent reference to Appendix A, especially in the earlier sections, to ease the transition to the required mathematical level for the subject. It is hoped that readers can use familiar topics from Appendix A as a springboard to less familiar topics in the text.

Finally, Appendix B lists a variety of references, for background and for further reading.

**Exercises.**
The text contains a number of exercises. Those very closely related to
textual material are inserted at the appropriate place. Additional
exercises are found at the end of each section, in a separate
subsection. I have tried to make the exercises thought provoking
without being too difficult. Hints are provided where appropriate.
Rather than always asking for computations or proofs, the exercises
sometimes ask for explanations and/or examples, to hopefully clarify the
subject matter in the student's mind.

**Prerequisites.**
As a prerequisite to reading this text, the
student should have a solid background
in basic undergraduate-level real analysis (*not* including measure
theory). In particular, the mathematical background summarised in
Appendix A should be very familiar. If it is not,
then books such as those in Subsection B.1 should be
studied first. It is also helpful, but not essential, to have seen some
undergraduate-level probability theory at the level of the books in
Subsection B.2.

**Further reading.**
For further reading beyond this text, the reader should examine the
similar but more advanced books of Subsection B.3.
To learn additional topics,
the reader should consult the books on pure
measure theory of Subsection B.4, and/or the advanced books
on stochastic processes of Subsection B.5, and/or the books
on mathematical finance of Subsection B.6. I would be
content to learn only that this text has inspired students to look at
more advanced treatments of the subject.

**Acknowledgements.**
I would like to thank several colleagues for encouraging me in this
direction, in particular Mike Evans, Andrey Feuerverger, Keith Knight,
Omiros Papaspiliopoulos,
Jeremy Quastel, Nancy Reid, and Gareth Roberts. Most importantly,
I would like to thank the many students who have studied these topics
with me; their questions, insights, and difficulties have been my main
source of inspiration.

Jeffrey S. Rosenthal

Toronto, Canada, 2000

http://probability.ca/jeff/

Contact me

**Second Printing (2003).**
For the second printing, a number of minor errors
have been corrected. Thanks to Tom Baird, Meng Du, Avery Fullerton,
Longhai Li, Hadas Moshonov, Nataliya Portman, and Idan Regev for helping
to find them.

**Third Printing (2005).**
A few more minor errors were corrected, with thanks to
Samuel Hikspoors, Bin Li, Mahdi Lotfinezhad, Ben Reason,
Jay Sheldon, and Zemei Yang.

- Many small additional topics have been added, and existing topics expanded. As a result, the second edition is over forty pages longer than the first.
- Many new exercises have been added, and some of the existing exercises have been improved or "cleaned up". There are now about 275 exercises in total (as compared with 150 in the first edition), ranging in difficulty from quite easy to fairly challenging, many with hints provided.
- Further details and explanations have been added in steps of proofs which previously caused confusion.
- Several of the longer proofs are now broken up into a number of lemmas, to more easily keep track of the different steps involved, and to allow for the possibility of skipping the most technical bits while retaining the proof's overall structure.
- A few proofs, which are required for mathematical completeness but which require advanced mathematics background and/or add little understanding, are now marked as "optional".
- Various interesting, but technical and inessential, results are presented as remarks or footnotes, to add information and context without interrupting the text's flow.
- The Extension Theorem now allows the original set function to be defined on a semialgebra rather than an algebra, thus simplifying its application and increasing understanding.
- Many minor edits and rewrites were made throughout the book to improve the clarity, accuracy, and readability.

Jeffrey S. Rosenthal

Toronto, Canada, 2006

http://probability.ca/jeff/

Contact me

**Second Printing (2007).**
A few very minor corrections were made, with thanks to
Joe Blitzstein and Emil Zeuthen.

Preface to the First Edition vii Preface to the Second Edition xi 1. The need for measure theory 1 1.1. Various kinds of random variables 1 1.2. The uniform distribution and non-measurable sets 2 1.3. Exercises 4 1.4. Section summary 5 2. Probability triples 7 2.1. Basic definition 7 2.2. Constructing probability triples 8 2.3. The Extension Theorem 10 2.4. Constructing the Uniform$[0,1]$ distribution 15 2.5. Extensions of the Extension Theorem 18 2.6. Coin tossing and other measures 21 2.7. Exercises 23 2.8. Section summary 27 3. Further probabilistic foundations 29 3.1. Random variables 29 3.2. Independence 31 3.3. Continuity of probabilities 33 3.4. Limit events 34 3.5. Tail fields 36 3.6. Exercises 38 3.7. Section summary 41 4. Expected values 43 4.1. Simple random variables 43 4.2. General non-negative random variables 45 4.3. Arbitrary random variables 49 4.4. The integration connection 50 4.5. Exercises 52 4.6. Section summary 55 5. Inequalities and convergence 57 5.1. Various inequalities 57 5.2. Convergence of random variables 58 5.3. Laws of large numbers 60 5.4. Eliminating the moment conditions 61 5.5. Exercises 65 5.6. Section summary 66 6. Distributions of random variables 67 6.1. Change of variable theorem 67 6.2. Examples of distributions 69 6.3. Exercises 71 6.4. Section summary 72 7. Stochastic processes and gambling games 73 7.1. A first existence theorem 73 7.2. Gambling and gambler's ruin 75 7.3. Gambling policies 77 7.4. Exercises 80 7.5. Section summary 81 8. Discrete Markov chains 83 8.1. A Markov chain existence theorem 85 8.2. Transience, recurrence, and irreducibility 86 8.3. Stationary distributions and convergence 89 8.4. Existence of stationary distributions 94 8.5. Exercises 98 8.6. Section summary 101 9. More probability theorems 103 9.1. Limit theorems 103 9.2. Differentiation of expectation 106 9.3. Moment generating functions and large deviations 107 9.4. Fubini's Theorem and convolution 110 9.5. Exercises 113 9.6. Section summary 115 10. Weak convergence 117 10.1. Equivalences of weak convergence 117 10.2. Connections to other convergence 119 10.3. Exercises 121 10.4. Section summary 122 11. Characteristic functions 125 11.1. The continuity theorem 126 11.2. The Central Limit Theorem 133 11.3. Generalisations of the Central Limit Theorem 135 11.4. Method of moments 137 11.5. Exercises 139 11.6. Section summary 142 12. Decomposition of probability laws 143 12.1. Lebesgue and Hahn decompositions 143 12.2. Decomposition with general measures 147 12.3. Exercises 148 12.4. Section summary 149 13. Conditional probability and expectation 151 13.1. Conditioning on a random variable 151 13.2. Conditioning on a sub-sigma-algebra 155 13.3. Conditional variance 157 13.4. Exercises 158 13.5. Section summary 160 14. Martingales 161 14.1. Stopping times 162 14.2. Martingale convergence 168 14.3. Maximal inequality 171 14.4. Exercises 173 14.5. Section summary 176 15. General stochastic processes 177 15.1. Kolmogorov Existence Theorem 177 15.2. Markov chains on general state spaces 179 15.3. Continuous-time Markov processes 182 15.4. Brownian motion as a limit 186 15.5. Existence of Brownian motion 188 15.6. Diffusions and stochastic integrals 190 15.7. Ito's Lemma 193 15.8. The Black-Scholes equation 194 15.9. Section summary 197 A. Mathematical Background 199 A.1. Sets and functions 199 A.2. Countable sets 200 A.3. Epsilons and Limits 202 A.4. Infimums and supremums 204 A.5. Equivalence relations 207 B. Bibliography 209 B.1. Background in real analysis 209 B.2. Undergraduate-level probability 209 B.3. Graduate-level probability 210 B.4. Pure measure theory 210 B.5. Stochastic processes 210 B.6. Mathematical finance 211 Index. 213