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Counterexamples in probability and real analysis. (English) Zbl 0827.26001
Oxford: Oxford University Press. xii, 211 p. (1993).
The term counterexample is used in this book as generally accepted in mathematics. The main goal is to clarify the role of conditions under which some statement is true or not. And as it turns out quite frequently the counterexamples do not correspond to our intuition. But in all the cases they can prevent us of falling into “gaps”.
The authors have compiled over 300 counterexamples. Most of them are in Real Analysis (Chapters 1-6). The remaining ones are in Probability, Statistics and some applications (Chapters 7-10).
In their content the counterexamples are very diverse. They range from elementary (but also useful) to quite complicated ones requiring special constructions and arguments. Several statements included in the book can be considered as counterexamples in a very generalized sense. It is obvious that the book represents the taste and the interests of the authors.
A natural question is: To whom in this book addressed? Perhaps the answer will not be unique. However, any analyst will find here a large number of nontrivial even surprising statements regarding the real line, differentiation, measure theory, and integration. Further, any stochastician will make use of the statements concerning probability, random processes, convergence, parameter estimation and decision theory. And some counterexamples are addressed to scientists in the applied fields where concepts like optimal estimation, detection and quantization are of great importance.
Having some experience in the field of counterexamples in probability and stochastic processes, the reviewer was glad to find new and interesting ones which appear for first time. Some of these new counterexamples are created by the authors themselves. Otherwise, the book is written in the usual style traditionally followed by the authors of books on counterexamples. There is no doubt that this book, together with other existing books of counterexamples in analysis, probability and statistics, will be useful for graduate students as well as for professional mathematicians.

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration