Synopses & Reviews
A counterexample is any example or result that is the opposite of one's intuition or to commonly held beliefs. Counterexamples can have great educational value in illuminating complex topics that are difficult to explain in a rigidly logical, written presentation. For example, ideas in mathematical sciences that might seem intuitively obvious may be proved incorrect with the use of a counterexample. This monograph concentrates on counterexamples for use at the intersection of probability and real analysis, which makes it unique among such treatments. The authors argue convincingly that probability theory cannot be separated from real analysis, and this book contains over 300 examples related to both the theory and application of mathematics. Many of the examples in this collection are new, and many old ones, previously buried in the literature, are now accessible for the first time. In contrast to several other collections, all of the examples in this book are completely self-contained--no details are passed off to obscure outside references. Students and theorists across fields as diverse as real analysis, probability, statistics, and engineering will want a copy of this book.
Review
"An interesting and informative book....The reader will find that it is stimulating and thought provoking, and lead to better understanding of real analysis and probability. Highly recommended for college and university libraries with graduate programs in mathematical sciences." --Choice
"The almost 350 examples from analysis, probability and statistics amount to a collection of interesting examples that may be useful in relevant courses on the subjects treated." --Journal of the American Statistical Association
"Will be of a great deal of interest to analysts and some probabilists." --Mathematical Reviews
Table of Contents
1. The Real Line
2. Real Valued Functions
3. Differentiation
4. Measures
5. Integration
6. Product Spaces
7. Basic Probability
8. Conditioning
9. Convergence in Probability
10. Applications of Probability