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Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces. (English) Zbl 0718.28008

Oxford Science Publication. Oxford: Clarendon Press. x, 168 p. £25.00 (1990).
This is a highly recommendable textbook giving a solid introduction to the ergodic theory of measurable dynamical systems with discrete time. The author essentially restricts attention to the simplest setting, that of an invertible bimeasurable measure-preserving transformation of a probability space. Under mild regularity assumptions, the probability space is isomorphic to the unit interval with Lebesgue measure, that is, it is a Lebesgue space. The central problem studied in the book is to find methods for classifying such dynamical systems up to isomorphy in the measurable sense.
As the author explains, the text is not intended to be encyclopaedic. Its aim is to present the most fruitful chains of arguments, notably the set coding combinatorial poit of view used by D. Ornstein and his school (to which the author belongs).
The prerequisite for reading the book is thorough working knowledge of basic concepts in set theoretic topology and measure theory, including the Riesz representation theorem.
The author starts with well chosen motivating examples, and with a few fundamental concepts. Chapter 2 offers a short treatment of the structure of Lebesgue space. [Most other books on ergodic theory delete this topic and refer instead to the long classical article of Rohlin.] An \(L_ 1\)- martingale theorem is proved via a backward Vitali type argument which is also used later in other proofs. Chapter 3 presents the simplest ergodic theorems and the ergodic decomposition of measure-preserving transformations. Chapter 4 develops the classical hierarchy of mixing concepts ending with Kolmogorov automorphisms. Chapter 5 introduces entropy via name counting, the Shannon-McMillan-Breiman theorem, the Pinsker algebra and related topics. Chapters 6 and 7 present the core of the modern theory on isomorphy. The concepts of joining and disjointness are presented. It is shown that Chacon’s transformation has minimal selfjoinings, and this is used to construct a number of counterexamples. Finally, Chapter 7 presents the Burton-Rothstein proofs of Krieger’s generator theorem and Ornstein’s isomorphism theorem.
The reading of the book is demanding, but it is rewarding since deep and useful insights are presented. The bibliography is rather short. For an introductory book, advice for further reading and some more historical information would have been helpful. (For example, it is not mentioned that the maximal ergodic theorem for measure-preserving transformations belongs to Yosida-Kakutani. The papers of Burton and Rothstein are not quoted.)
This book is an excellent source of information on modern ergodic theory; for those who work on problems related to isomorphy of measurable dynamical systems, it is a must.

MSC:

28D10 One-parameter continuous families of measure-preserving transformations
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration