This is the only book that gives a rigorous and comprehensive treatment with lots of examples, exercises, remarks on this particular level between the standard first undergraduate course and the first graduate course based on measure theory.
There is no competitor to this book.
The book can be used in classrooms as well as for self-study.
From the reviews of the second edition:
This is an excellent introductory book on random variables, with a wealth of examples and exercises. ... The material is very well organized ... . The text is remarkably well written, mathematically and aesthetically; layout and fonts make it a pleasant reading, and the examples are often enlightening. I think it will be a valuable support for students and instructors and it should definitely find a place in every good library. (Fabio Mainardi, The Mathematical Association of America, October, 2009)
...A worthwhile addition to the textbook pool, one that will guide the student safely through to a point of competence and ability to embark on a more advanced study... (International Statistical Review, 2010, 78, 1, 134-159)
The book addresses a unique niche mathematically inclined students previously exposed to an introductory course in probability ... . The writing style is lucid and easy to follow. ... book is clearly directed toward mathematicians and the highly mathematically inclined scientist or engineer who might be induced to study the mathematics of probability or mathematical statistics. For those who find the classical mathematical pedagogy motivating or those requiring a comprehensive readable reference work on the mathematics of probability theory the book can be highly recommended. (Thomas D. Sandry, Technometrics, Vol. 53 (1), February, 2011)
This book ... is intended as an introductory graduate level textbook in probability for statistics majors. ... This book provides an elaborate description and a collection of results in probability theory. ... The level of this book is suitable for a graduate course. Overall all concepts are well discussed with full mathematical rigor. ... has a good collection of most of the results related to probability theory, the price is very reasonable, and I will recommend this book to university mathematics and statistics libraries. (Sounak Chakraborty, Journal of the American Statistical Association, Vol. 106 (495), September, 2011)