**1. Bases mathématiques du calcul des probabilités (1964), by Jacques Neveu.**

**1.1. Review by: S Orey.**

*Mathematical Reviews*MR0198504

**(33 #6659)**.

This slim volume is designed to serve students of mathematics as an introduction to probability theory. The chapter headings are as follows: (I) Probability spaces; (II) Integration of random variables; (III) Product spaces and random functions; (IV) Conditional expectations and martingales; (V) Ergodic theory and Markov processes. The presentation is concise throughout, and the text is supplemented with important exercises. The final chapter is particularly useful in that it collects and organizes several important topics from the research literature.

**1.2. Review by: D G Kendall.**

*The Mathematical Gazette* **50** (371) (1966), 82-83.

MM Bourbaki have never written a treatise on probability theory; had they done so, it might have looked like this. Neveu's book will be widely welcomed as an important contribution to the literature, and many pure mathematicians whose interest in the subject was sparked off by the London Mathematical Society's Durham Conference will find here the "further reading" they have been lacking.

**2. Mathematical foundations of the calculus of probability (1965), by Jacques Neveu.**

**2.1. Comment.**

English translation of *Bases mathématiques du calcul des probabilités *(1960).

**2.2. From the Publisher.**

The aim of this book is to develop the mathematical foundations of the theory of random processes to the point where the reader should have no difficulty in further pursuing the subject in any direction he chooses. To this end, the author develops the theory of measure and integration ab initio; the presentation is distinguished by a high degree of elegance and polish, as well as by the depth and completeness of the exposition. Then general random processes are taken up, and Doob's results on separability and measureability of random processes are proved. The theory of martingales and submartingales follows; here the basic convergence and continuity properties of such processes are proved. The next subject is Markov processes (discrete time) with general state spaces. The general theory is presented, followed by ergodic theory for such processes. Both the mean and individual ergodic theorems are proved. the latter in the general form given by Chacon and Ornstein. Approximately 100 highly interesting and far-ranging problems are included. It should serve admirably for a course in advanced probability theory, and as a textbook for a high-level course on measure theory. It will provide material for self-study to anyone who has had an elementary course in probability theory.

**2.3. From the Preface.**

The object of the theory of probability is the mathematical analysis of the notion of chance. As a mathematical discipline, it can only develop in a rigorous manner if it is based upon a system of precise definitions and axioms. Historically, the formulation of such a mathematical basis and the mathematical elaboration of the theory goes back to the 1930's. In fact, it was only at this period that the theory of measure and of integration on general spaces was sufficiently developed to furnish the theory of probability with its fundamental definitions, as well as its most powerful tool for development.

Since then, numerous probabilistic investigations, undertaken in the theoretical as well as practical domain, in particular those making use of functional spaces, have only served to confirm the close relations established between probability theory and measure theory. These relations are, incidentally, so close that certain authors have been loath to see in probability theory more than an extension (but how important a one!) of measure theory.

In any case, it is impossible at the present time to undertake a profound study of probability theory and mathematical statistics without continually making use of measure theory, unless one limits oneself to a study of very elementary probabilistic models and, in particular, cuts oneself off from the consideration of random functions. Attempts have been made, it is true, to treat convergence problems of probability theory within the restricted framework of the study of distribution functions; but this procedure only gives a false simplification of the question and further conceals the intuitive basis of these problems.

The book represents the essentials of a course for the first year of the third cycle (which corresponds roughly to the first or second year of graduate work in the United States) which is addressed to students who already have some elementary notions of the calculus of probability; it is intended to furnish them with a solid mathematical base for probability theory. Only a reader with a sound mathematical development could consider this book an introduction to the theory of probability.

Our first aim in this course is therefore to teach the reader how to handle the powerful tools provided by measure theory and to permit him subsequently to deal with any chapter of probability theory. Numerous problems complement the text; given the very "technical" nature of the subject being treated, it would seem to us indispensable for the reader to try to read and solve the greater part of these problems. (To help the reader in this task, we have frequently sketched a solution of a problem.) In accordance with a presently well-established French tradition concerning introductory treatises, we have not deemed it worthwhile to insert bibliographical references in the text, or, with rare exceptions, to attribute the results obtained to their various authors. The reader will find, at the end of the book, a concise bibliography relating to the text or to the complements; most of the problems, in particular, arise out of the works listed in this bibliography.

We would not wish to conceal from the reader the fact that measure theory is not the unique tool of probability theory, even though it is its principal tool; we could not too strongly advise him to learn, if he has not already done so, the precise notions of topology, the theory of metric spaces, and the theory of Hilbert and Banach spaces. This book could not contain within its limited confines any introduction to these theories. Certain problems, and even certain portions of the text, [we have marked them with an asterisk] make use of notions borrowed from these theories; the beginner can ignore them without fear of losing the thread of the presentation, while the more advanced reader will be able to find connections with outside fields which may interest him.

I wish to take this opportunity to thank Professors R Fortet, M Loève and A Tortrat for their suggestions and encouragement. The form of this book also owes much to the reactions of the students who have taken my course. Finally, my thanks go equally to Dr A Feinstein for his excellent work of translation.

**2.4. From the Introduction.**

The fundamental concepts of the theory of probability are those of events and of probabilities: Axiomatically, events are mathematical entities which are susceptible of combination by the logical operations "not," "and," "or" (according to the rules specified in Section 1 of this chapter), while a probability is a valuation on the class of events whose properties are by definition analogous to those of a frequency (see Section 3).

Another notion, which is in fact frequently introduced as the first notion of the theory of probability, is that of a trial, that is, the results a random experiment. From the natural condition of considering only events and trials relating to the experiment which is being studied, every trial necessarily determines, by its definition, either the realization or non-realization of every event which one wishes to consider. We are thus led to introduce the ensemble Ω of trials (or possible results of the experiment being considered) and to identify each event with the subensemble of trials which realize this event; a probability thus becomes a set function, similar to a volume defined on certain subsets of a Euclidean space. The preceding ensemble point of view is that of measure theory, which we shall develop in the first chapter. With regard to probabilities, we have defined them first on Boolean algebras (or, as in Section 6, on Boolean semialgebras), following which we extend them to *σ*-algebras and thus construct probability spaces. This procedure has the advantage of exhibiting a very important extension theorem of measure theory; moreover, in the construction of probabilities on Euclidean spaces or on product spaces (see Chapter 3), probabilities turn out to be defined naturally, at the outset, on algebras or semialgebras.

**2.5. Review by: F Eicker.**

*SIAM Review* **9** (1) (1967), 136-139.

The book presents a rigorous measure-theoretic introduction to the calculus of probability. It employs throughout the abstract terminology and methods of relevant branches of modern mathematics. As a consequence, the book resembles in style and emphasis very much a monograph in "pure" mathematics. The material is condensed almost to the absolutely necessary minimum from a logical point of view. Thus the utmost of conciseness and of elegance has been achieved, yet the presentation remains clear and transparent. On the whole the book might well be considered to exemplify a new style in the writing of graduate texts in the area of probability theory. Another example is the recent little book by H Bauer [Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie]. Because of its high mathematical standards not everybody will find the book easy reading. Accordingly, the book is concerned only very little with the phenomenological background, applications in practice, or detailed motivation. No examples interrupt the development. Particular prerequisites are acquaintance with the elements of topology, functional analysis and the theory of metric, Hilbert and Banach spaces (only some details on *L*_{p}-spaces are presented). The reference to Dunford and Schwartz's monograph on linear operators, whose terminology is used, may prove to be quite helpful. In the reviewer's opinion, however, it would have been feasible and advantageous to include the material actually needed from these disciplines in the book.

**2.6. Review by: R M Blumenthal.**

*The Annals of Mathematical Statistic*s **38** (2) (1967), 624-625.

In a relatively small amount of space this book presents the part of measure theory most relevant to study and research in modem probability theory, and also presents the most important parts of ergodic theory and martingale theory. The presentation is elegant, but with a reasonable allowance for human frailty. Anyone intending to learn out of this book must be able to read graduate-level mathematics. Also it will be helpful if he has some intuitive background from an elementary course in probability or statistics. If he is this well-equipped, he will find the study of this book an enjoyable and profitable experience. Having mastered the material (supplemented by a bit of extra work on characteristic functions) he will be prepared for reading most of the current research papers in probability as well as many of those in mathematical statistics. The book is suitable for use as a course textbook or for individual study. There are problems at the end of each section, usually containing important information. The problems are not short or easy, but the most interesting ones are broken up into short steps, so that most serious readers will be able, ultimately, to solve them.

**2.7. Review by: J F C Kingman.**

*Journal of the Royal Statistical Society. Series A (General)* **129** (3) (1966), 475-476.

It is becoming increasingly common to assert (or to admit) that the mathematical theory of probability is just a branch of the theory of measure and integration. Although it would perhaps be more accurate to describe measure theory as an amusing generalization of the simpler aspects of probability theory, there is no doubt that the formalism of measures on general spaces does provide a more satisfactory framework for the manipulation of random variables and processes than do the earlier and vaguer ideas. It is this framework to which Professor Neveu has provided a very competent introduction. In approach it resembles Loève's masterpiece *Probability Theory*, although the result is a very different book, more modest and perhaps more useful. It might be described as a "Child's Guide to Loève", so long as it is understood that the child in question must possess a very fair degree of mathematical maturity as well as a good grounding in elementary probability theory. The author begins by developing, in a clear and fairly standard way, the necessary results on measure and integration, which he does using the language and special conditions of probability theory, thereby achieving a useful simplification. He goes on to introduce random functions (i.e. stochastic processes) and gives a nice treatment of the Daniell- Kolmogorov theorem and its extension by Ionescu Tulcea. These ideas are then applied to martingales and, at some length, to ergodic theorems for stochastic (and especially Markov) processes. ... too elementary or too advanced. This book has a good deal to offer a reader who knows something about the subject already and will fill in the gaps from other sources; both provisos are necessary. The translation is accurate if sometimes inelegant.

**2.8. Review by: M Csörgö.**

*Canadian Math. Bull. ***10** (2) (1967), 308.

This is an excellent book by a brilliant mathematician, expertly and beautifully translated into English. As a text it is intended for study by advanced students (corresponding roughly to the first or second year of graduate work in probability or measure theory in the United States and Canada) and as a reference work for researchers. There is no doubt that it will succeed brilliantly in fulfilling these aims.

The book develops the mathematical foundations of the theory of random processes to the point where the reader should have no difficulty in further pursuing the subject in any direction he chooses. To this end, the theory of measure and integration is developed including all the theorems for construction of a probability measure by extension from an algebra to a

Approximately 100 interesting and far-ranging problems are included and thus even decision theory and sufficient statistics are treated to some extent.

Summarizing, this excellent book should serve admirably for a course in advanced probability theory, and as a textbook for a high level course on measure theory. It is also delightful reading material for anyone interested in this field and can be also used for self-study purposes by one who has had an elementary course in probability or measure theory.

**3. Processus aléatoires gaussiens (1968), by Jacques Neveu.**

**3.1. Review by: M Loève.**

*Mathematical Reviews*MR0272042

**(42 #6923).**

A successful attempt of the author to present a first synthesis of recent developments in the theory of Gaussian processes. The first half is devoted to Gaussian spaces associated to a random function or a Gaussian measure and their isomorphisms with spaces *L*^{2} and spaces of self-reproducing kernels. It consists of four chapters: (I) Gaussian random variables, (II) Gaussian spaces, (III) Gaussian random functions, and (IV) Gaussian random measures. Chapter V is a basis for the prediction problems for vector-valued stationary random functions. It is entitled: "Gaussian random measures defined on Hilbertian integrals. Canonical representation of Gaussian random functions." Chapter VI, entitled "Tensor products of Hilbert spaces", investigates symmetric tensor powers of a Hilbert space, and this concept finds its applications in the next two Chapters, VII and VIII, on Wiener chaos and on equivalence of Gaussian processes. ... It is to be hoped that the promise by the author of a definitive book on this important subject will be realized in the not too distant future.

**4. Bases mathématiques du calcul des probabilités (2nd edition) (1970), by Jacques Neveu.**

**4.1. From the Preface.**

This second edition differs essentially from the first 1964 edition only by the addition of a paragraph at the end of Chapter IV ('Independent Random Variable Sequences'). Improvements which have already benefited from the English translations [1965], Russian [1969] and German [1969] have been made to the text, and of course I have also corrected the errors that various readers have been kind enough to report to me.

**5. Martingales à temps discret (1972), by Jacques Neveu.**

**5.1. Review by: D L Burkholder.**

*Mathematical Reviews*MR0402914

**(53 #6728)**.

This is an elegant exposition of discrete-time martingale theory. Although the restriction to discrete time eliminates some of the most exciting applications, the discrete-time case offers many fundamental challenges and gives rise to most of the key ideas of martingale theory. The first chapter introduces the reader, who is assumed to have a working knowledge of measure and integration theory and some acquaintance with probability theory, to the definition and basic properties of conditional expectations. ... The second chapter begins with a discussion of stopping times. Positive supermartingales are then defined and studied. ... In the third chapter, devoted to applications, the Lebesgue decomposition theorem for measures and the Radon-Nikodým theorem are derived using martingale theory. ... In Chapter IV, the author presents Doob's submartingale convergence theorem and related inequalities. ... The fifth chapter contains material on martingales indexed by arbitrary directed sets, vector-valued martingales, and reversed martingales. The next chapter is devoted to the kind of optimization problem that arose in Wald's theory of sequential statistical analysis and which was studied in greater generality by Snell and, later, by Chow, Robbins, Siegmund, Dynkin and others. ... The last two chapters of the book are devoted to a "first synthesis'' of some theoretical results that have been obtained during the last few years. Many of these results are stated in terms of increasing processes and potentials, ideas of great importance in continuous-time martingale theory. ... Overall, this is a valuable addition to the growing number of excellent books on martingale theory.

**5.2. Review by: C Doléans-Dade.**

*SIAM Review* **16** (3) (1974), 398-399.

This book is a very clear and well written presentation of almost all classical and recent results on discrete martingales. The reader is assumed to have a good knowledge of integration theory and to be familiar with elementary probability theory. Some of the proofs given in this book are due to the author. These are often much simpler than the original ones. The author has also been careful to point out, in numerous remarks, the traps into which the reader might fall. These remarks are certainly very helpful for any one not familiar with martingale theory.

**6. Discrete-parameter martingales (1975), by Jacques Neveu.**

**6.1. Comment.**

English translation of *Martingales à temps discret* (1972).

**6.2. From the Preface.**

This volume presents a third cycle one-semester course first taught at the Université de Paris during 1970-71. It was published in French in 1972 by Masson et Cie. The present translation differs from the French text mainly in its last two chapters which have been rewritten in order to take into account the recent progresses of the theory.

Without any doubt the theory of martingales constitutes the mathematical technique at the base of modern probability. We have restricted ourselves to "discrete-time" martingales and give an exposition which, we hope, is fairly complete. At the outset the reader is assumed to be familiar with integration theory and to possess the basic notions of probability theory as taught in mathematics courses.

The first chapter gives a deep study of conditional expectations, without assuming of the reader any previous knowledge on this topic. Chapters II and IV then contain the basic results concerning discrete-time martingales, essentially the convergence and stopping theorems. Chapter III contains important applications, mainly to measure theory and Markov chains; Chapter V presents the various extensions of the notion of martingale (martingales with directed index set, vector-valued martingales, reversed martingales). Chapter VI is entirely devoted to an optimisation problem which has been of interest to numerous mathematicians over recent years. To finish up, Chapters VII and VIII present important theoretical results obtained in the last few years, and constitutes a first attempt at a synthesis of this material. Finally, a lengthy bibliography testifies to the intense research activity taking place in the theory of martinga1es; with some exceptions I have only cited those works on discrete-time martingales which have appeared since 1960.

I would like to thank my colleagues and students for their numerous remarks concerning the original text. I would also like to give particular thanks to Madame Baltzer and Mademoiselle Gillet for their excellent work in preparing the text for print. It is equally pleasant for me to be able to thank Dr T P Speed for the care, patience and competence with which he carried out the translation and revision of the French text.

**6.3. Review by: D A Darling.**

*Bull. Amer. Math. Soc.* **82** (6) (1976), 836-840.

The present work seems to be the first devoted exclusively to the study of martingales, which the author says "without any doubt constitute the mathematical technique at the base of modern probability". It is doubtful that a statement of such scope would find general agreement, and it is a fact that few authors of books on probability have given an extensive treatment. In the preface Neveu states that the material in the book has been used as a one semester course in the "third cycle" at the University of Paris. This corresponds roughly to the second year of graduate studies in American universities. A good course in measure theory is a prerequisite, as is a knowledge of the "basic notions of probability theory". In addition the development requires a knowledge of some aspects of linear space theory (including basic Hilbert space theory), although such things as Holder's inequality are proved. ... Thus a well-prepared and motivated second year graduate student could undertake this material, but to finish it in one semester would require intensive work. ... The exposition is first rate, as in earlier works of the author. It is clean, economical and elegant, with admirable directness in presenting the right theorems at the right places. It is well motivated without overselling the subject and what is often rare in the writings of French probabilists, it is accessible to the general public. It is a great pleasure to read casually or study intensively.