1.1. From the Preface.

Ever since Yu V Prohorov wrote his fundamental paper "Convergence of random processes and limit theorems in the theory of probability" in the year 1956, the general theory of stochastic processes has come to be regarded as the theory of probability measures in complete separable metric spaces. The subsequent work of A V Skorohod of which a detailed account is given in the book by I I Gikhman and A V Skorohod gave a great impetus to the subject of limit theorems in probability theory. The present monograph deals with the general theory of probability measures in abstract metric spaces, complete separable metric groups, locally compact abelian groups, Hilbert spaces, and the spaces of continuous functions and functions with discontinuities of the first kind only. Emphasis is given to the work done by the Indian School of probabilists consisting of, among others, V S Varadarajan, R Ranga Rao, S R S Varadhan, and the author.

In Chapter I a detailed description of the Borel $\sigma$-field of a metric space is given. The isomorphism theorem which states that two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality is proved. The by now classical theorem of Kuratowski on the measurability of the inverse of a measurable map from a Borel set of a complete separable metric space into a separable metric space is established.

Chapter II deals with properties such as regularity, tightness, and perfectness of measures defined on metric spaces. The weak topology in the space of probability measures is investigated and Prohorov's theorems on metrisability and compactness are proved. Convergence of sample distributions and existence of nonatomic measures in separable metric spaces are investigated.

Chapters III and IV deal with the arithmetic of probability distributions in topological groups. The nature of idempotent measures, existence of indecomposable and nonatomic indecomposable measures, and representation of infinitely divisible distributions and its relevance in the theory of limit theorems for "sums" of infinitesimal summands are discussed in great detail.

Chapter V contains the proofs of the classical extension theorems and existence of conditional and regular conditional probabilities in standard Borel spaces. This is included for the sake of completeness.

Chapter VI gives a detailed account of the subject of limit theorems in a Hilbert space. Prohorov's theorems on the description of weakly compact subsets of the space of measures on a Hilbert space, Sazonov's criterion for a characteristic function, and Varadhan's results on the Lévy-Khinchine representation of infinitely divisible laws and limit theorems for sums of infinitesimal summands are established.

Chapter VII is devoted to a study of probability measures on the spaces $C[0, 1]$ and $D[0, 1]$. Properties of the Skorohod topology in the space $D[0, 1]$ are proved. The compactness criteria for sets of probability measures and their applications to testing statistical hypotheses are given.

The account of the subject given herein is by no means complete. The topic of measures on Lie groups and homogeneous spaces, for example, is not touched on at all. Some of the references on this subject have been included, however, and for further comments the reader may refer to the bibliographical notes at the end of the book.

Theorem numbers refer to theorems in the same chapter if they are not accompanied by a chapter number.

The author wishes to express his thanks to R Ranga Rao, V S Varadarajan, and S R S Varadhan with whom he had the opportunity of collaboration over several years. It is a pleasure to offer thanks to F Topsøe and O Bjornsson who read the first two chapters of the manuscript carefully and made valuable suggestions. He is grateful to V Rohatgi and K Vijayan who also helped with proofreading.

He is deeply indebted to the Research and Training School of the Indian Statistical Institute, Calcutta, for providing facilities to carry out research work during 1960-1965, a substantial portion of which has been included in the present monograph. He is grateful to the Department of Probability and Statistics of the University of Sheffield for the invaluable assistance rendered in the preparation of the book.

Thanks are finally due to Miss J Bowden for her efficient typing of the manuscript.

1.2. Contents.

Chapter 1. The Borel subsets of a metric space.
Chapter 2. Probability measures in a metric space.
Chapter 3. Probability measures in a metric group.
Chapter 4. Probability measures in locally compact abelian groups.
Chapter 5. The Kolmogorov consistency theorem and conditional probability.
Chapter 6. Probability measures in a Hilbert space.
Chapter 7. Probability measures on $C[0, 1]$ and $D[0. 1$].

1.3. Review by: R M Dudley.
The Annals of Mathematical Statistics 40 (1) (1969), 328.

This book seems to have been written just at the right time, when most of its subject matter has reached the appropriate degree of maturity.

Nearly half the book is concerned with generalising the Lévy-Khinchin representation of infinitely divisible probability measures and associated facts to locally compact abelian groups (Chapter 4) and to Hilbert space (Chapter 6). The author himself and other members of the Indian school, along with V Sazonov, are responsible for most of the results.

Chapter 3 gives preliminaries about measures on metric groups. Chapter 5 studies inverse limits of Borel spaces and regular conditional probabilities, and Chapter 7 studies probabilities on $C[0, 1]$ and $D[0, 1]$.

Chapters 1 and 2 present material on Borel sets and probability measures in separable metric spaces which is useful as a prerequisite not only to Chapters 3-7 but for wide areas of probability and functional analysis; more useful, the reviewer feels, than the study of non-separable locally compact spaces in Halmos' classic Measure Theory. The research worker needing further information can consult V S Varadarajan's monograph "Measures on topological spaces."

The book under review contains no exercises, but it is carefully and cogently done and should be quite readable for graduate students.

1.4. Review by: B L S Prakasa Rao.
Journal of the American Statistical Association 63 (323) (1968), 1061-1062.

A large part of classical probability theory consists of the study of limiting behaviour of normed sums of independent random variables. The study of classical theorems like the central limit theorem and the laws of large numbers in more general abstract spaces is not only of mathematical interest but is necessary for a number of practical problems in physics, biology, economics, etc.

The present monograph deals with the theory of probability measures in abstract metric spaces, complete separable metric groups, locally compact abelian groups, Hilbert spaces, and the spaces of continuous functions and functions with the discontinuities of the first kind only. The author says in the preface, "Emphasis is given to the work done by the Indian school of probabilists consisting of, among others, V S Varadarajan, R Ranga Rao, S R S Varadhan and the author" and it is indeed so.

Chapter 1 deals with theorems on Borel subsets of a metric space. Chapter 2 deals with regular measures, tight measures, and perfect measures on metric spaces. The weak topology in the space of probability measures is investigated and the theorems on compactness and metrisability of Prohorov are proved. In Chapters 3 and 4, probability measures on topological groups are discussed and the Levy-Khinchin representation for infinitely divisible distributions on locally compact abelian groups is obtained. Chapter 5 gives a lucid description of Kolmogorov's consistency theorem and the existence of conditional probability in Standard Borel spaces. Probability measures on Hilbert spaces are discussed in Chapter 6. Chapter 7 is devoted to a study of probability measures on the spaces $C[0,1]$ and $D[0,1]$. The compactness criteria for sets of probability measures and their applications to hypothesis testing are given.

The book is extremely well written and I would recommend it for any serious student in probability or mathematical statistics. It is well known that the results in Chapter 7 are used by mathematical statisticians in obtaining asymptotic results in nonparametrics as well as in problems of estimation and testing. It would have been useful for students in statistics if the author had included some recent applications of these results. There are very few misprints in the book. The author has to be congratulated for his excellent book which will develop further interest in the subject.

1.5. Review by: Ramesh A Gangolli.
Mathematical Reviews MR0226684 (37 #2271).

It has been known for some time that a fruitful way of studying the general theory of stochastic processes is to regard it as substantially equivalent to the theory of probability measures on complete separable metric spaces. The present book is devoted to giving a cohesive account of the latter theory. Much of the work reported on is original with the author and his collaborators, notably, V S Varadarajan, R Ranga Rao and S R S Varadhan. The result is a very readable book which should serve as an excellent source from which a student could learn the subject and also a convenient reference for the specialist for theorems which must by now be regarded as basic to the subject.

After giving a detailed description of the structure of Borel sets of a complete separable metric space, the author goes on to discuss the weak topology for probability measures, and treats regularity, tightness and perfectness of measures. There is included a discussion of convergence of sample distributions and also a number of metrisability and compactness theorems in the space of probability measures on a complete separable metric space.

Chapters III and IV are devoted to a discussion of the arithmetic of probability measures on metric groups and locally compact abelian groups. Notions such as decomposability, infinite divisibility, idempotence, and their relevance to limit theorems for "sums" of infinitesimal random variables are discussed in detail.

The remainder of the book is primarily devoted to the subject of limit theorems for probability measures on Hilbert space (Chapter VI) and on the spaces $C[0,1]$ and $D[0,1]$ (Chapter VII). Results of Sazonov-Gross, Varadhan and Skorohod on these topics are elaborated and their applications are briefly discussed, among them, theorems of Kolmogorov-Smirnov.

1.6. Review by: Thomas M. Liggett.
Technometrics 11 (4) (1969), 839-840.

This book represents a very useful contribution to the literature of probability theory. Much of the material presented is made available here for the first time in book form. The presentation is generally concise, but usually quite clear and understandable. The work is, as a whole, well written and well organised.

The first chapter deals with some of the technical properties of Borel subsets of a metric space which are needed throughout the book. The principal results obtained are the Isomorphism Theorem for Borel sets, and the Kuratowski Theorem on the measurability of the inverse of a one-one measurable map from one Borel subset of a complete separable metric space onto another. Chapter 2 studies the space of probability measures on a general metric space and includes, for example, Prohorov's Theorem on the weak relative compactness of sets of probability measures.

The next two chapters develop the theory of probability measures on locally compact Abelian groups. The main objective here is to generalise the classical results on infinitely divisible distributions on the line to the context of groups. In Chapter 4, the Pontrjagin duality theory and much of the structure theory for locally compact Abelian groups is assumed.

In Chapter 5, a rather general formulation of the Kolmogorov Consistency Theorem is presented, and the existence of regular conditional probabilities is studied. Chapter 6 concerns probability measures on Hilbert spaces. Again, the primary objective is a study of infinitely divisible distributions on a Hilbert space. Ergodic theorems for Hilbert space-valued random variables also are presented here.

The final chapter gives a rather brief description of the spaces of probability measures on $C[0, 1]$ and $D[0, 1]$, together with some standard applications of weak convergence in these spaces. The brevity of this chapter is justified by the recent publication of "Convergence of Probability Measures" by Patrick Billingsley, which studies these spaces in greater detail.

Professor Parthasarathy has written a very lucid account of much of the theory of measures on metric spaces. It should be very useful to students and researchers in the field.

1.7. Review by: E E Granirer.
Canadian Mathematical Bulletin 13 (2) (1970, 290-291.

From the appearance of Yu V Prohorov's fundamental paper "Convergence of random processes and limit theorems in the theory of probability" in 1956 much of the theory of stochastic processes has been regarded as the theory of probability measures on metric spaces.

The present book deals with the theory of probability measures in abstract metric space, complete metric groups, Hilbert spaces, spaces of continuous functions, etc. A survey of the contents of this book will clarify its scope:

Chapter I: The Borel subsets of a metric space. The Borel $\sigma$-field of a metric space is studied in detail. The basic isomorphism theorem which states that if $A, B$ are Borel subsets of complete separable metric space $X, Y$ and if $A, B$ have the same cardinality then there is a one to one bimeasurable map from $A$ onto $B$ is proved. Furthermore Kuratowski's theorem which asserts that if $\varphi$ is a one-to-one measurable map from a Borel subset $A$ of a complete separable metric (c.s.m.) space onto a Borel subset $B$ of another space then $\varphi^{-1}$ is also measurable, i.e. $\varphi$ is an isomorphism.

In Chapter II properties such as regularity tightness and perfectness of measures are defined and studied. Connections between measures and linear functionals and the weak convergence of measures are studied. Prohorov's theorems on compactness and metrisability of a set of measures are proved. Existence of nonatomic measures in separable metric spaces is investigated. The interesting theorem of the author, Ranga Rao, and Varadarajan which asserts that any uncountable complete separable metric $X$ space admits a nonatomic measure is proved. Its proof is beautiful.

In Chapters III and IV probability distributions in topological groups are studied in detail. A beautiful elementary proof due to the author, of the fact that any idempotent measure on a separable complete metric group is necessarily the Haar measure on a compact subgroup, is given. Representations of infinitely divisible measures in locally compact abelian groups and the connected limit theorems are brought.

Most of the results of these chapters are taken from the fundamental paper of the author, Ranga Rao, and V S Varadarajan, Probability distributions on locally compact abelian groups, Ill. (1963). The reader will, though, have to fill out quite a few details for himself in these chapters.

Chapter V entitled the Kolmogorov Consistency Theorem and Conditional Probability contains the proofs of the by now classical extension theorems and the existence of conditional and regular conditional probability in standard Borel spaces.

Chapters VI and VII deal with probability measures in a Hilbert space, in $C[0, 1]$ and in $D[0, 1]$ (functions possessing right and left limits at every point in $[0, 1]$. Limit theorems in Hilbert space, weakly compact sets of measures on Hilbert spaces (Prohorov theorems), Varadan's results on the Levy Khinchin representation of infinitely divisible laws, etc., are proved.

Probability measures on $C[0, 1]$ and $D[0, 1]$ are studied. The Skorohod topology in $D[0, 1]$ and its properties are proved. Compactness criteria for sets of probability measures and some applications to testing statistical hypotheses are given.

The book is self-contained and elegantly written. It is recommended reading for any graduate student or worker in the field of probability.

2.1. From the Introduction.

In the mathematical formulation of quantum mechanics proposed by Von Neumann the set of all propositions concerning a quantum mechanical system is an ortho-complemented lattice. The simplest example of such a lattice is the lattice $\mathfrak{L}(\mathfrak{H})$ of all closed subspaces of a separable Hilbert space $\mathfrak{H}$. The observables in such a system turn out to be self adjoint operators in $\mathfrak{H}$. In order to construct the standard observables like energy, linear, angular and spin angular momenta, etc., in such a system it is necessary to study the effect of coordinate transformations by a group $G$ of symmetries. In this context the study of representations of $G$ in the group of automorphisms of the lattice $\mathfrak{L}(\mathfrak{H})$ is of great importance.

2.2. Review by: G W Mackey.
Mathematical Reviews MR0246995 (40 #264).

Let $G$ be a locally compact group. The multipliers of the author are Borel functions $\sigma$ from $G \times G$ to the complex numbers of modulus one such that $\sigma(g_{1}, g_{2}g_{3})\sigma(g_{2}, g_{3}) = \sigma(g_{1}, g_{2})\sigma(g_{1}g_{2}, g_{3})$ and $\sigma(g, e) = \sigma(e, g) = 1$ for all $g_{1}, g_{2}, g_{3}, g \in G$. This article is a unified exposition of results about multipliers due to V Bargmann, the reviewer, V S Varadarajan and D J Simms in which, according to the introduction, "many of the proofs are simpler and some of the results are more general".

3.1. From the Preface.

These notes are mainly based on a course of lectures given by the first named author at the Research and Training School of the Indian Statistical Institute, Calcutta during May 1971. A first and slightly shorter version of these notes has appeared under the same title as publication No. M71-1 of the Research and Training School of the Indian Statistical Institute. Some of the results were obtained when the authors were at the Statistical Laboratory, Mathematics Department, University of Manchester, in 1970.

The notion of a continuous tensor product of Hilbert spaces and group representations appears in the work of H Araki, H Araki and E J Woods, and R F Streater. Their analysis leads to a connection between continuous tensor products and the theory of infinitely divisible distributions of Probability theory. The present work contains a systematic study of these notions in terms of positive definite kernels with invariance properties under a group action. Such analysis also leads to a unified approach to the central limit problems of Probability theory, the theory of stochastic processes with stationary increments and construction of free fields in Quantum Mechanics.

The contents of these notes are divided into three parts. In part 1 the notion of a projectively invariant positive definite kernel on an abstract $G$-space is introduced and obtained as expectation value of a projective unitary representation of the group $G$. Affine invariant conditionally positive definite kernels are investigated and a representation of such kernels in terms of unitary representations and first order cocycles is obtained. Using these ideas and the theory of multiplicative measures, continuous tensor products of Hilbert spaces and representations are constructed. The Fock-Cook construction of the Bose-Einstein field in Quantum Mechanics arises as a natural consequence of this theory. Much of the inspiration for this approach was derived from the work of H Araki.

Part 2 analysis limits of products of uniformly infinitesimal families of positive definite kernels. This leads in particular to the limit laws in the theory of sums of independent random variables.

The third part contains results about first order cocycles for various classes of representations and on some special groups. In particular it is shown that first order cocycles of induced representations arise from the cocycles of the inducing representation. As corollary to this, cocycles of irreducible representations of nilpotent Lie groups are described. Cocycles of representations of semisimple Lie groups which are induced by characters of maximal solvable subgroups are also obtained.

A short list of references including most of the major contributions to these problems is included.

The first named author would like to thank the Research and Training School of the Indian Statistical Institute and the University of Bombay for providing him facilities for writing these notes. For the same reason the second author would like to express his gratitude to the Department of Mathematics, Bedford College, University of London.

3.2. Review by: Terence P Speed.
Mathematical Reviews MR0622034 (58 #29849).

A positive definite (p.d.) kernel on a space $X$ is a map $K: X \times X \rightarrow \mathbb{C}$ such that for any finite subset $\{x_{1,}x_{2}, ... , x_{n}\} \subseteq X$ the matrix $(K(x_{i},x_{j}))$ is nonnegative definite. A map $L: X \times X \rightarrow \mathbb{C}$ is said to be conditionally positive definite (c.p.d.) if $exp(t L)$ is p.d. for all $t > 0$. The major part of these notes concerns the case where $X$ is a $G$-space for some group $G$, and the kernel $K$ transforms under $G$ in some specified manner. ... Part I of these notes, entitled "Positive definite kernels and continuous tensor products", begins by discussing the relations between p.i.p.d. kernels, projective representations of groups, and cocycles, together with their affine analogues: affine invariant c.p.d. kernels and additive cocycles. ... Part I closes with a brief mention of the relationship of this to Bose-Einstein fields. - The second part of the notes, "Limit theorems for uniformly infinitesimal families of positive definite kernels", is a very interesting attempt (not yet complete) to derive the central limit theorems of probability in the more general framework of p.d. kernels. ... The final part, "The analysis of cocycles of the first order on some special groups", uses the machinery of group representation theory to determine explicitly the first-order cocycles on locally compact abelian, compact, nilpotent Lie and semisimple Lie groups.

4.1. From the Preface.

In 1902 the French mathematician Henri Lebesgue wrote his famous dissertation Intégrale, Longueur, Aire (Integral, Length and Area). Since 1914 the theory of the Lebesgue measure has become a part of the undergraduate curriculum in analysis in all the technologically advanced countries of the world. In 1933 the Russian mathematician A N Kolmogorov wrote the famous book Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of Probability) in which he formulated the basic axioms of probability theory. The appearance of Measure Theory by P R Halmos and An Introduction to Probability Theory and Its Applications by W Feller in 1950 made both subjects accessible to all undergraduate and graduate students in mathematics all over the world.

The present book has been written in the hope that it will provide the impetus to introduce in undergraduate and graduate programmes both measure theory and probability theory as a one-year course. Since the study of probability theory in its advanced stage depends on a knowledge of measure theory, special effort has been made to integrate the two subjects into a single volume.

The material of the book grew out of the lectures delivered by the author to M.Sc. students at the Centre of Advanced Study in mathematics in the University of Bombay, M.Stat. students of the Indian Statistical Institute, Delhi, and M.Sc. students at the Indian Institute of Technology, Delhi.

The book is divided into eight chapters. Chapter 1 deals with combinatorial probability and the classical limit theorems of Poisson and Laplace-De Moivre. It provides a motivation for extending measures from boolean algebras to $\sigma$-algebras. Chapter 2 is devoted to extension of measures from boolean semialgebras and classes of topologically important subsets to $\sigma$-algebras. Chapter III deals with properties of borel maps from a measure space into a separable metric space. In particular, Lusin's theorem and the isomorphism theorem are proved. Extension of measures to projective limits of borel spaces is also studied. Chapter 4 deals with integration, Reisz's representation theorem that integration is the only linear operation on good function spaces, and properties of function spaces arising out of a measure space.. Chapter 5 contains a discussion of measures and transition measures on product spaces. The Lebesgue measure in $\mathbb{R}^{k}$, the change of variable formula for Lebesgue integrals and construction of infinitely differentiable functions are also considered. Chapter 6, which is the longest in the book, introduces the notion of conditional expectation through orthogonal projection and avoids the customary use of the Radon-Nikodym theorem. The Radon-Nikodym theorem and the Lebesgue decomposition are deduced as a corollary of a more general decomposition theorem due to von Neumann. Convergence of conditional expectations in various senses, the idea of regular conditional probability, ergodic theorems and ergodic decomposition are also treated in this chapter. Chapter 7 gives a brief introduction to weak convergence of probability measures and characteristic functions. The last chapter introduces the construction of Haar measure on a locally compact group and invariant and quasi invariant measures on homogenous spaces. The Mackey-Weil theorem on groups with a quasi invariant measure is also proved. For the benefit of the student a number of exercises are included. Connections between measure theory and probability theory on the one hand and various topics like functional analysis, statistics, ergodic theory, etc., on the other are indicated through Remarks, Examples and Exercises.

4.2. Review by: L E Clarke.
The Mathematical Gazette 63 (426) (1979), 281-282.

This book may be briefly described as a course on general measure theory with special reference to probability (and a second course rather than an introduction). Although much of the text is traditional, the treatment and choice of topics often reflect Professor Parthasarathy's own interest in measures on metric spaces, and there are some pleasing detours from time-honoured paths.

In the opening chapter the author concerns himself solely with finitely additive probabilities, but even here he establishes the central limit theorem and the strong law of large numbers for simple random variables. In the next 150 pages he covers the general theory of measure, measurable functions and integration, including measures on product spaces and Fubini's theorem. He follows this with a short introduction to Hilbert space theory, which is used to establish the existence of a conditional expectation and to prove the Radon-Nikodym theorem. He continues with the weak convergence of probability measures and characteristic functions, and concludes with a short chapter on invariant measures on groups.

The above summary may give some idea of the multitude of topics treated, even though it has made no reference to such interesting digressions as that from Hilbert space theory via the Parseval identity to a solution of the isoperimetric problem. However, this profusion is only achieved at the price of a treatment which is often so condensed as to make the book unsuitable, in my opinion, for undergraduates and even for first-year graduates, and any student user would require more exercises than those provided (about 140).

The book would be much improved by better paper and typography (it was printed in Madras) and fewer misprints, though those that occur are rarely serious.

There is much to be commended in this text, and it is a pity that Professor Parthasarathy has not been able, or allowed by his publisher, to develop his main themes in a more leisurely manner.

4.3. Review by: Marius Iosifescu.
International Statistical Review / Revue Internationale de Statistique 47 (2)
(1979), 233-234.

This book integrating into a single volume basic elements of both measure theory and probability theory for a one-year course is divided into 8 chapters as follows:

1. Probability on Boolean algebras (comprising much of the elementary probability theory);
2. Extensions of measures;
3. Borel maps (including, in particular, Lusin's theorem, the isomorphism theorem, and extensions of measures to projective limits of Borel spaces);
4. Integration (including Riesz's representation theorem)
5. Measures on product spaces (not mentioning Ionescu Tulcea's theorem which clearly should be invoked in Exercise 35.21);
6. Hilbert space and conditional expectation (avoiding the customary use of Radon-Nikodym theorem);
7. Weak convergence of probability measures (including Prohorov's theorem);
8. Invariant measures on groups (including Mackey-Weil theorem on groups with a quasi-invariant measure).

Many helpful exercises, remarks, and examples are included in all the chapters.

As a minor criticism this reviewer does not agree with the dropping of the capital letter in 'Boolean' and 'Borel' (used as an adjective, e.g. 'a borel map').

The author was able to treat very clearly in a small number of pages a great amount of selected material much of which is not yet usual for introductory books. Of course, to achieve this, his experience of an outstanding research mathematician has been instrumental.

Written in the hope to provide an impetus to introduce probability theory and measure theory in the Indian undergraduate and graduate programmes, the present book is destined to render invaluable services to the international mathematical community on both teaching and research sides.

4.4. Review by: Terence P Speed.
Mathematical Reviews MR0651013 (58 #31322b).

This book was originally written to provide Indian mathematics students at the M.Sc. level with a text which introduces them to probability theory, at the same time providing the measure-theoretic background necessary for a more advanced study of this subject. Consequently, most of the material is standard and the treatment familiar, but there are a number of aspects of this well-written book which should be of interest to teachers in the field, as well as to the students at whom it is aimed. The eight chapter headings are as follows: I. Probability on Boolean algebras; II. Extensions of measures; III. Borel maps; IV. Integration; V. Measures on product spaces; VI. Hilbert space and conditional expectation; VII. Weak convergence of probability measures; VIII. Invariant measures on groups.

The book provides a careful well-motivated explanation of the basic measure theory and its application to mathematical probability. Concepts are introduced, briefly discussed, illustrated by examples, and then the main results concerning them are proved. In this way the basic material of the first five chapters is rapidly and interestingly covered. Some exercises are interspersed with the text, but in general these serve the purpose of illustration, completion or minor extension of the preceding discussion. Points of interest in the first half of the book include the early introduction of measures on metric spaces, with an associated section on probability contents, and an unusually full discussion of the much-used change-of-variable formula involving the Jacobian. This last result is then used in the derivation of Liouville's theorem on the preservation of Lebesgue measure in Euclidean phase-space by a one-parameter group of homeomorphisms.

The sixth and by far the longest chapter begins with an introduction to Hilbert space and the basic theory concerning orthogonal projections. Parseval's theorem is illustrated in a Fourier-analytic solution of the isoperimetric problem, and a good discussion of orthogonal polynomials is given, all before conditional expectations are introduced. A rapid review of the elementary facts concerning conditional expectations includes the easiest form of the martingale convergence theorem, but with no mention of martingales. The basic result giving regular conditional probabilities, the Radon-Nikodým and Lebesgue theorems concerning measures, the strong law of large numbers and some ergodic theorems close this chapter.

In the next chapter, we find the first results concerning weak convergence and Prohorov's theorem and then turn to a discussion of Fourier transforms on $\mathbb{R}^{k}$. The final chapter gives the construction of Haar measure on locally compact second countable metric groups, a useful formula for invariant measures on subsets of Euclidean space and a discussion of quasi-invariant measures on homogeneous spaces. It ends with a proof of the Mackey-Weil theorem asserting that a standard Borel space with a quasi-invariant measure $mu possesses a locally compact topology whose Borel <latex>\sigma$-algebra coincides with the given one, and whose Haar measure is equivalent to $\mu$.

From the foregoing it can be seen that the book provides a good coverage of certain basic topics, as well as containing a fair quantity of nonstandard material and a point of view which gives it a character of its own. These all make it a worthwhile book, yet one cannot help but feel that something is lacking. It seems to be that the book is slightly old-fashioned in its style and content, and whilst this is not in itself a criticism, it becomes relevant when one considers the nature of probability theory in the late 1970's and the fact that the book aims at providing an introduction to this material. The tremendous advances in stochastic processes and limit theorems in recent years could be reflected in many ways in an introductory text, for example, through use of the more effective Dynkin systems in the early stages of introducing and extending measures, rather than of semi-algebras and monotone classes, or an emphasis on martingales as a basic tool in probability theory, or in a discussion of conditional independence and the Markov property. In a sense this book only introduces the reader to the "static" aspects of probability theory such as the theory of infinite-divisibility, rather than the more "dynamic" ones associated with the behaviour of stochastic processes.

Summarising, this is an interesting and readable text, with an emphasis which, although not as modern as this reviewer would like, promises to catch the imagination of many readers and direct them towards further study. One cannot ask much more of such a book.

4.5. Review by: J D Biggins.
Journal of the Royal Statistical Society. Series D (The Statistician) 28 (3)
(1979), 230.

This book provides a thorough introduction to measure theory and to the more measure-theoretic side of probability theory. The measure theory covers, as one would expect, such topics as the extension of measures from algebras of sets to $\sigma$-algebras, measurable mappings, integration theory and product spaces. The probability theory treated includes conditioning, the ergodic theorem and weak convergence. At times the book takes on a distinct functional-analytic flavour, for example in the discussion of $L_{p}$ spaces.

The high level of abstraction makes the introductory nature of the book less than apparent at times. An illustration is that Prohorov's theorem is proved without any preliminary discussion of the case when the probability measures are all concentrated on the real line. I suspect that one would not wish to use this book as an introductory text, despite its title.

The mathematical background needed, a knowledge of general topology, is not always covered in undergraduate mathematics courses, particularly not those with a sizeable probability and statistic content. This would seem to limit the usefulness of the book as a teaching text. There are many typographical errors and though most are easy to spot they might well cause difficulty for the unwary.

4.6. Review by: J F C Kingman.
Journal of the Royal Statistical Society. Series A (General) 142 (3) (1979), 385-386.

There was a time when the connection between probability and abstract measure theory was an arcane fact, of interest to mathematicians at the research level but not, for instance, informing undergraduate teaching. Then in the sixties came textbooks aimed at an undergraduate audience, and spelling out the ways in which measure theory gives a firm basis for the calculus of probability, with its many important applications. Now such books are common, and a new arrival tends to be greeted with muted enthusiasm.

Parthasarathy has written a textbook in this mould, but it is welcome both because it is a scholarly yet readable account, and because it differs from others in choice of material and emphasis of treatment. Probability and measure are developed in parallel from first principles, with the essential results on integrals, distributions and mappings. The distinctive features are all reflections of the author's research interests. For example, random variables are allowed to take values in metric spaces, and weak convergence is treated in this natural setting. Conditional expectation is tied more closely to Hilbert spaces than is commonly the case. An interesting final chapter concerns probability measures on groups, though the opportunity is lost to introduce the reader to geometrical probability.

This then is a definite addition to the literature. It would be a good accompaniment to a course at undergraduate or graduate level in which the teacher could supplement Parthasarathy's mathematics with more account of applications. And certainly the specialist will find much to stimulate thought in this refreshing personal view of what the author thinks most important in the subject.

5.1. Review by: B M Makarov.
Mathematical Reviews MR0506158 (80d:47001).

This is a text for students containing the following four chapters: I. Banach spaces. II. Linear operators. III. Analytic operator-valued mappings. IV. Calculus in $B$-spaces.

The first two chapters contain a survey of standard facts of the theory of Banach spaces and operators in them. Many statements are listed without proof, and a number of secondary propositions are found in the exercises. The presentation is illustrated with examples. Applications of the general results to ordinary differential equations are discussed in detail.

Chapter III contains less traditional material. Here analytic operator-valued functions are studied, in particular the question of the analogue of a Jordan form for such a function, projector-valued functions, the analyticity of eigenvalues. In the first two sections mainly the finite-dimensional case is considered, while Section 3 is devoted to operators in infinite-dimensional Banach spaces.

The fourth chapter contains elements of the theory of smooth mappings in Banach spaces (Fréchet differential, Taylor formula, implicit functions, absolute and relative extrema).

6.1. Note.

The 1977 version of this book (see [] above) was published by Macmillan in Delhi. The 1978 version was published by Springer-Verlag in New York.

7.1. From the Preface.

This is a continuation of Lectures on functional analysis, Part I. It is based on lectures delivered by the author at the Indian Statistical Institute (New Delhi) and the Tata Institute of Fundamental Research (Bangalore) during the academic year 1977-78.

The contents are divided into two chapters, V and VI. Chapter V is devoted to a detailed study of the Kato topology in the space of closed operators on Banach spaces and the stability properties of closedness, invertibility, nullity, deficiency, index, spectrum and essential spectrum of an operator under different kinds of perturbations. Chapter VI deals with operators in Hilbert space. It contains a proof of the Kato-Rellich criterion for selfadjointness of a symmetric operator and its applications in the study of Schrödinger and Dirac operators. A detailed discussion of the spectra of Schrödinger operators of atoms with one or two electrons is included.

7.2. Review by: D E Edmunds.
Bulletin of the London Mathematical Society 13 (3) (1981), 254.

While Part I dealt with bounded perturbations, unbounded operators come to the fore in this short volume, which consists of only two chapters, V and VI. Chapter V discusses closed linear operators acting from one Banach space to another and is concerned with the behaviour, under different kinds of perturbation, of closedness, invertibility, the index, the spectrum, and the essential spectrum; particular attention is paid to Fredholm and semi-Fredholm maps. Hilbert space topics are handled in Chapter VI, and here we find the Kato-Rellich criterion for self-adjointness of a symmetric operator, along with applications to the Schrödinger and Dirac operators.

Most of the material in these two chapters can be found in Chapters 4 and 5 of T. Kato's classic book, "Perturbation Theory", and those looking for substantial originality or novelty of proof will be disappointed, as for the most part the author has been content to follow Kato very closely. The material is presented clearly and accurately, but my principal feeling after reading it is one of regret tinged with surprise that Professor Parthasarathy should have chosen to publish an account of the subject so exactly modelled upon Kato's 1966 book without taking real advantage of the opportunity to incorporate more of the significant advances that have been made in the intervening years. Those who have Kato's book will not really need this more recent volume; those still without it might do better to read the master's account rather than the second-hand version provided by the book under review

8.1. From the Preface.

In the physicist's picture of quantum theory the notion of a real valued observable has, like Brahma, three visible faces: (i) a spectral measure on the line; (ii) a selfadjoint operator in a Hilbert space; (iii) a unitary representation of the real line as an additive group. The equivalence of these three descriptions is a consequence of von Neumann's spectral theorem for a, not necessarily bounded, selfadjoint operator and Stone's theorem on the infinitesimal generator of a one parameter unitary group in a Hilbert space. Real valued random variables on a classical probability space $(\Omega , \mathfrak{F}, P)$ when viewed as selfadjoint multiplication operators in the Hilbert space $L^{2}(P)$ are special examples in the quantum description. This suggests the possibility of developing a theory of quantum probability within the framework of operators and group representations in a Hilbert space. The first chapter of the present work is aimed at an exploration of this idea assuming only a knowledge of elementary functional analysis and classical probability.

Through the papers of R F Streater in 1969, H Araki in 1970 and K R Parthasarathy and K Schmidt in 1972 there has emerged a string of ideas exhibiting the close relationship between the infinitely divisible probability distributions of Levy and Khinchin on the one hand and the projective unitary Weyl representation of the Euclidean group of a Hilbert space in the associated symmetric (boson) Fock space on the other. The Weyl representation is a neat summary of the Weyl commutation relations or, equivalently, the canonical commutation relations (CCR) and the second quantisation homomorphism. In its infinitesimal form the Weyl representation leads naturally to the fundamental notions of creation, conservation and annihilation operators which obey the (extended) CCR and can therefore be interpreted as the source of Heisenberg's uncertainty principle. Classical stochastic processes with independent increments are then realised as distributions in the vacuum state of commuting families of observables which are, essentially, linear combinations of the creation, conservation and annihilation operators in the backdrop of a filtration determined by a time observable. Such a realisation depends very much on the theory of tensor products of Hilbert spaces and operators on them. The reader may find a fairly self-contained approach to this theme in the second chapter.

In classical probability theory, infinitely divisible distributions are realised from stochastic processes with independent increments of which the paradigm examples are the standard Brownian motion and the Poisson process of a given intensity. Starting with the pioneering efforts of N Wiener, K Ito, P Levy and J L Doob there has now evolved during the last six decades an extraordinarily rich stochastic calculus around these and more general local semimartingales encompassing stochastic integration, Ito's formula, stochastic differential equations and a variety of applications which constitute the fruit bearing orchard of modern stochastic analysis. (See, for example, the book of N Ikeda and S Watanabe). The developments in the second chapter indicate the possibility of developing a quantum stochastic calculus around the basic creation, conservation and annihilation operators arising from the Weyl representation of the Euclidean group of a Hilbert space equipped with a continuous time observable in order to induce a filtration. The motivation for such an attempt lies in the intuitive feeling that the description of any dynamical phenomenon depends on the creation of some objects, their preservation and motion for a period of time followed by their annihilation and, finally, the repetition of such a process in cycles, all subject to the laws of chance. A theory of quantum stochastic differential equations based on the notions of creation, conservation and annihilation operator-valued processes and time may possibly help in examining the point of view expressed above. Starting with the preliminary ideas in, an attempt to develop a quantum stochastic calculus along the lines mentioned already was made in 1984 by R L Hudson and K R Parthasarathy. The principal aim of the third chapter is to present such a calculus leading to quantum Ito's formula and highlight the following essential features: (i) the classical Ito's formula for Brownian motion and Poisson process can also be viewed as consequences of CCR and hence the Heisenberg's uncertainty principle; (ii) canonical anticommutation relations (CAR) and CCR can be derived from each other through stochastic integration; (iii) quantum dynamical semigroups describing irreversible evolutions result from averaging the solutions of a Heisenberg equation in the presence of noise, strengthening the belief that irreversible evolutions are most likely to be shadows of reversible evolutions in an enlarged universe; (iv) quantum stochastic flows (in the sense of Evans-Hudson) arise from quantum stochastic differential equations; (v) classical Markov chains can be realised in the quantum framework through stochastic difference and differential equations for observables with coefficients depending on appropriate group actions.

The list of topics omitted and reasons for their omission is too long to be enumerated and hence the number of friends and colleagues to whom I must apologise is very high. Primarily, the subject is very young and dynamic and the growth of literature in recent years is so explosive that I can hardly keep pace with it. Secondly, notions like conditional expectation, stop time, local time etc., are in the process of assuming a visible shape but the body of interesting physical as well as mathematical examples seems to require much more nourishment. There do exist several other approaches to quantum stochastic integration like the one based on kernels in Guichardet's symmetric measure space due to H Maassen and the other on $C^{*}$- and $W^{*}$-algebraic methods due to C Barnett R F Streater and I F Wilde and L Accardi and F Fagnola. The interested reader may find access to the voluminous literature on all these recent developments in the Lecture Notes of proceedings of the workshops in quantum probability and its applications edited by L Accardi, W von Waldenfels and others as well as the expositions of P A Meyer entitled "Eléments de probabilités quantiques" in the Séminaire de Probabilités.

The present work is a revised version of the mimeographed notes which owes its existence, primarily, to the influence of R L Hudson with whom the author had the pleasure of collaboration during the last nine years. Indeed, this monograph was first proposed to be written jointly with him but the occurrence of such a happy event was prevented by the difficulties involved in simultaneously occupying the states of being the chairman of a department in a British university during hard times and indulging in the luxury of organising already discovered results. To R L Hudson and his colleagues D Applebaum and J M Lindsay at Nottingham, P A Meyer at Strasbourg and L Accardi at Rome, I express my heartfelt gratitude for uncountable conversations and exchanges of ideas on the subject of quantum stochastic calculus as well as the nonmeasurable hospitality I enjoyed during my visits to their departments and homes; to R F Streater who showed and emphasised passionately that Fock space is a fertile soil for new developments in probability theory; to my friend and colleague K B Sinha for his enthusiastic collaboration and illuminating suggestions whenever analytical difficulties obstructed imagination; to my devoted audience of three consisting of S K Muthu, A Mohari and B V R Bhat during my year-long seminar at the Delhi Centre of the Indian Statistical Institute during 1988; and to V S Varadarajan, the guru who initiated me to the charms of the geometry of quantum theory through his marvellous lectures on G W Mackey's approach to the mathematical foundations of quantum mechanics during the year 1965 in Calcutta prior to his total exit from the Indian mathematical scene. I am deeply indebted to B V R Bhat, J M Lindsay and S K Muthu in checking the mushroom growth of mistakes in my manuscript as well as suggesting ways for its improvement. Special thanks are due to V P Sharma for his patient and elegant preparation of the manuscript in TEX. Finally, no amount of thanks would suffice for the cheerful and encouraging support I have received from Shyama on the domestic front where life in Delhi in the sweltering summer heat of forty plus, accompanied by aandhi winds is an everyday battle right from facing an erratic power and water supply to procuring the daily necessities of life like bread, rice, vegetables and milk.

8.2. Review by: David Applebaum.
Mathematical Reviews MR1164866 (93g:81062).

Quantum stochastic calculus is an operator-theoretic generalisation of the well-known classical stochastic calculus, initiated by K Itô in the 1940s, for stochastic processes which are driven by Brownian motion or the Poisson process. It is motivated by the need to be able to describe the irreversible dynamical behaviour of quantum systems interacting with noise and the "intuitive feeling that the description of any dynamical phenomenon depends on the creation of some objects, their preservation and motion for a period of time followed by their annihilation and, finally, the repetition of such a process in cycles, all subject to the laws of chance" (page x). Although the subject is rapidly developing in several different directions, its central core is the content of a series of papers, published in the early 1980s by the author and R L Hudson, which originated the theory and which are an essential prerequisite to understanding current research in the field. The aim of the present volume is to give a concise account of these results and of some of their more important consequences.

The book is divided into three main sections. The first part is introductory and consists mainly of well-known material organised from the novel perspective of doing probability theory on the lattice of projections in a Hilbert space. The central idea is the concept of a "random variable" (i.e. quantum observable) in this context and its equivalent mathematical descriptions as a spectral measure, selfadjoint operator (via the spectral theorem) and unitary representation of the real line (via Stone's theorem). "Probability distributions" of such random variables are completely determined by Gleason's theorem. An essential observation is that two quantum random variables need not commute and hence cannot be simultaneously realised as (multiplication operators by) classical random variables in the same probability space. Hence the need arises for a theory of "quantum probability".

The second section gives a thorough treatment of Hilbert space tensor products and the Fock spaces (free, bosonic and fermionic) which are the natural home for our subject. The main emphasis is on the boson case which has so far been the most popular and fruitful of the three. The Weyl representation of the Euclidean group of a Hilbert space is introduced and its infinitesimal form (the extended canonical commutation relations or CCR) described in terms of creation, conservation and annihilation operators. As motivation for the next section it is shown that any infinitely divisible random variable can be represented in a suitable boson Fock space by means of these operators.

The final section describes quantum stochastic calculus based on the creation, conservation and annihilation processes in Fock space. Stochastic integrals of adapted processes are constructed and the vitally important quantum Itô formula is proved via the extended CCR. These are then applied to construct a representation of the canonical anticommutation relations in boson Fock space. Quantum stochastic differential equations with bounded coefficients are considered and necessary and sufficient conditions obtained for their solution to be unitary. Conjugation by these latter processes is the prototype for quantum stochastic flows of Evans-Hudson type, and furthermore yields dilations of the most general form of uniformly continuous quantum dynamical semigroup. The author gives the Mohari-Sinha construction of a more general class of Evans-Hudson flows which are driven by noise with infinitely many degrees of freedom, applications of which include an elegant construction of classical Markov chains in Fock space.

This is an excellent volume which will be a valuable companion both to those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students. I have only two small criticisms. Firstly the introductory section could have been considerably shortened without great loss as much of the material therein is readily available elsewhere. Secondly the use of generalised time observables (rather than Lebesgue measure on $\mathbb{R}^{+}$ in the account of quantum stochastic calculus seems unnecessarily complicated and yields little gain to the reader.

8.3. Review by: Paul-André Meyer.
Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 56 (1) (1994), 174-175.

Quantum Probability means probability theory developed under von Neumann's axioms rather than the usual (Kolmogorov's) axioms. Instead of being measurable functions on a measure space, random variables are self-adjoint operators on a Hilbert space, and probability laws are defined by state vectors or, more generally, density matrices. Non-commutativity of operators prevents random variables from having joint distributions.

A given (classical) probabilistic situations $(\Omega , \mathfrak{F}, \mathfrak{P})$ can be extended into a quantum probabilistic one as follows: the basic Hilbert space is $\mathfrak{H} = L^{2}(\mathfrak{P})$; the probability law is the state vector 1 (the function 1), and with each classical (real valued) random variable $X$ is associated the operator of multiplication by $X$ on $L^{2}$, which is self-adjoint, unbounded if $X$ is unbounded in the classical sense. This universal method gives rise to interesting quantum probabilistic problems if one is able to single out natural families of self-adjoint operators which do not commute with the multiplication operators. This is the case in ordinary quantum mechanics, which is usually realised on the Hilbert space $L^{2} (\mathfrak{R}>^{n})$, but which can also be realised on $L^{2}(\gamma)$, the standard Gaussian distribution on $\mathfrak{R}>^{n}$. Then the position operators $Q_{i}$ appear as multiplication operators, and the momentum operators $P_{i}$ are natural self-adjoint operators which do not commute with the preceding ones.

Quantum Stochastic Calculus arises when this finite dimensional Gaussian measure is replaced by the infinite dimensional Wiener measure of the Brownian motion process $(B_{t})$. Then the multiplication operator by the r.v. $B_{t}$ (usually denoted by $Q_{t}$) can be split into the sum $a_{t} + a_{t}^{*}$ of two non commuting operators, the annihilation and creation operators, and the family of operators $P_{t} = (a_{t}^{*} - a_{t})$ can be interpreted as arising from multiplication by another Brownian motion. These two families, however, do not commute together, but instead satisfy canonical commutation relations. We recover in this way the quantum Brownian motion, defined and studied long ago by R L Hudson.

This is only the starting point of the story. In an epoch making paper, Hudson and Parthasarathy extended to this non-commutative set-up the Ito theory of stochastic integration with respect to Brownian motion (non commutative integrals had been introduced at about the same time by Barnett, Streater and Wilde for fermionic Brownian motion), and started extending to the non-commutative case the theory of stochastic differential equations. In the same paper, they also demonstrated how the number operator process $(\Lambda _{t})$ arises on the same footing as the creation and annihilation processes as a "stochastic integrator". Including the number process permits the non-commutative unification of the two classical theories of Brownian motion and Poisson processes.

Other major steps were the unification of the bosonic and fermionic Brownian motions as Jordan-Wigner transforms of each other, and the development of the theory of "quantum stochastic flows".

All this, and much more, is included into this masterly exposition of the theory, which takes the subject at its very beginning, and develops it in complete detail, adding also many fundamental theorems from quantum mechanics (Von Neumann's spectral theorem, Gleason's theorem, Wigner's theorem on the automorphisms of the state space of a Hilbert space, Stinespring's theorem on completely positive maps...). It also contains a large number of exercises and examples.

9.1. From the Publisher.

Having been out of print for over 10 years, the AMS is delighted to bring this classic volume back to the mathematical community. With this fine exposition, the author gives a cohesive account of the theory of probability measures on complete metric spaces (which he views as an alternative approach to the general theory of stochastic processes). After a general description of the basics of topology on the set of measures, he discusses regularity, tightness, and perfectness of measures, properties of sampling distributions, and metrisability and compactness theorems. Next, he describes arithmetic properties of probability measures on metric groups and locally compact abelian groups. Covered in detail are notions such as decomposability, infinite divisibility, idempotence, and their relevance to limit theorems for 'sums' of infinitesimal random variables. The book concludes with numerous results related to limit theorems for probability measures on Hilbert spaces and on the spaces $C[0, 1]$. "The Mathematical Reviews" comments about the original edition of this book are as true today as they were in 1967. It remains a compelling work and a priceless resource for learning about the theory of probability measures. The volume is suitable for graduate students and researchers interested in probability and stochastic processes and would make an ideal supplementary reading or independent study text.

10.1. From the Preface.

This is a brief introduction to the mathematical foundations of quantum mechanics based on lectures given by the author to Ph.D. students at the Delhi Centre of the Indian Statistical Institute during the years 1980-1985 in order to initiate active research in the emerging field of quantum probability. The material in the first chapter is included in the author's book "An Introduction to Quantum Stochastic Calculus" published by Birkhauser Verlag in 1992 and the permission of the publishers to reprint it here is acknowledged. Apart from quantum probability, an understanding of the role of group representations in the development of quantum mechanics is always a fascinating theme for mathematicians. In this context the books by G W Mackey and V S Varadarajan have exerted a considerable influence in my pedagogical approach to the subject. Graduate students to whom I had recommended these books felt somewhat uncomfortable with them and I hope these notes would prove to be more encouraging in getting a quicker introduction to this theme.

The first chapter deals with the definitions of states, observables and automorphisms of a quantum system through Gleason's theorem, Hahn-Hellinger theorem and Wigner's theorem. Mackey's imprimitivity theorem and the theorem of inducing representations of groups in stages are proved directly for projective unitary antiunitary representations in the second chapter. Based on a discussion of multipliers on locally compact groups in the third chapter all the well-known observables of classical quantum theory like linear momenta, orbital and spin angular momenta, kinetic and potential energies, gauge operators etc., are derived solely from Galilean covariance in the last chapter. A very short account of observables concerning a relativistic free particle is included. In conclusion, the spectral theory of Schrödinger operators of one and two electron atoms is discussed in some detail.

I have benefited greatly from discussions with R Bhatia, M Krishna, P L Muthuramalingam and K B Sinha in writing these notes. The first three participated actively in the preparation of an earlier version and correcting innumerable mistakes. To all of them I express my sincere thanks and I take the responsibility for all the surviving errors. Finally, I thank M Krishna for revising the notes, organising the TEXing and making it possible to bring out this edition in the form of a lecture note volume.

I am grateful to Mr V P Sharma for typing the original lecture notes and Mrs T S Bagya Lakshmi for typing a part of the current version. Generous support from the Institute of Mathematical Sciences, Chennai in the completion of the revised version during the period 19 June - 31 July 2005 is gratefully acknowledged.

10.2. Review by: Pavel B Kurasov.
Mathematical Reviews MR2188339 (2007g:81001).

This book provides a self-contained introduction into mathematical methods of quantum mechanics with the emphasis on group representations. It is much influenced by the classical book by H Weyl [The theory of groups and quantum mechanics, 1949].

The book consists of four chapters. The first chapter is reprinted from the book by the same author [An introduction to quantum stochastic calculus, 1992] and contains proofs of Gleason's and Wigner's theorems. The first theorem associates to any state on a Hilbert space a unique non-negative operator with unit trace. The second theorem states that every automorphism of the set of states is given by either a unitary or an antiunitary operator. The second chapter is devoted to systems with a configuration invariant under a group action with the emphasis on imprimitivity systems. It is shown that in construction of such systems projective unitary-antiunitary representations play an important role. Multipliers on different locally compact groups are considered in the third chapter. The last chapter presents concrete expressions for certain standard observables like position, momentum, etc.

The book can be used as a basis for an interesting course on mathematical foundations of quantum mechanics. One has to take into account that prospective students should be prepared by taking courses in group representations and introductory quantum mechanics. It may be recommended to extend the contents by using the classical monograph by J von Neumann or the excellent book by L Faddeev and O Yakubovskiĭ (existing in Russian only so far).

Finally it is hard to resist copying here the citation from The Yajurveda to be found on the cover: "As are the crests on the heads of peacocks, as are the gems on the hoods of cobras, so is mathematics, at the top of all sciences."

11.1. From the Preface.

Probability theory had a humble origin during the middle of the seventeenth century in the study of games of chance whereas measure theory was born in the quest to understand Fourier series in the beginning of the twentieth century. To borrow a phrase attributed to Mark Kac and its adaptation by my friend Luigi Accardi, probability theory and quantum probability are respectively measure and operator theory with a soul. This sentiment is well brought out in A N Kolmogorov's "Foundations of Probability" (1933) and von Neumann's "Mathematical Foundations of Quantum Mechanics" (1932). A recognition of this aesthetic aspect depends very much on a good understanding of measure theory. Owing to recent explosive developments in probability theory from the view point of applications there is a similar need among a wide spectrum of scholars ranging from economists to engineers and physicists to psychologists. Furthermore, measure theory has its ramifications in topics like function spaces, operator theory, generalised functions, ergodic theory, group representation, quantum probability etc.

Taking all the aspects mentioned above into our view the manuscript for the 1977 edition of the book was prepared by using the notes based on the M.Sc. and M.Stat courses which I taught at the University of Bombay, I.I.T., Delhi and the Delhi Centre of the Indian Statistical Institute. It was typed on an ancient mechanical typewriter and printed by a publisher, not very much accustomed to the delicate needs of a subject like mathematics. However, the main aim of bringing out an inexpensive local edition and making the study of probability and measure occupy a visible position in our graduate and undergraduate programmes was achieved. Unfortunately, the book disappeared from the market around 1982. Students who took courses based on this book have communicated their appreciation on several occasions and also expressed their disappointment at its nonavailability in the market. The present, albeit infinitesimally revised, edition with several corrections has been skilfully TEXed by Anil Shukla and printed by the most competent professional publisher in India in the field of mathematics and, I believe, it would go a long way in helping students get a firm foothold in the twin themes of probability and measure and understand the sentiments expressed above. To Anil Shukla and the Hindustan Publisher I offer my sincere thanks.

12.1. From the Preface.

These notes were prepared by Amitava Bhattacharya on a course of lectures I gave at the Tata Institute of Fundamental Research (Mumbai) in the months of April 2001 and February 2002. I am grateful to my colleagues at the TIFR, in general, and Professor Parimala Raman in particular, for providing me a receptive and enthusiastic audience and showering on me their warm hospitality. I thank Professor Jaikumar for his valuable criticism and insight and several fruitful conversations in enhancing my understanding of the subject. Finally, I express my warm appreciation of the tremendous effort put in by Amitava Bhattacharya for the preparation of these notes and their LATEX files.

Financial support from the Indian National Science Academy (in the form of C V Raman Professorship), TIFR (Mumbai) and Indian Statistical Institute (Delhi Centre) is gratefully acknowledged.

13.1. From the Preface.

The logarithmic connection between entropy and probability was first enunciated by L E Boltzmann (1844-1906) in his kinetic theory of gases. His famous formula for entropy $S$ is S =$k \log W$ (as engraved on his tombstone in Vienna) where $k$ is a constant and $W$ is the number of possible microstates corresponding to the macroscopic state of a system of particles in a gas. Ignoring the constant $k$ and replacing $\log W$ by $-\log P(E)$ where $P(E)$ is the probability of an event $E$ in the probability space $(\Omega , \mathfrak{F}, P)$ of a statistical experiment, C E Shannon (1916-2001) looked upon $-\log P(E)$ as a measure of the information gained about the probability space from the occurrence of $E$. If $X$ is a simple random variable on this probability space assuming the values $a_{1}, a_{2}, ..., a_{k}$ from a finite set with $P(X = a_{j}) = p_{j}$ for each $j$ then the famous Shannon entropy $H(X) = - \Sigma _{j}p_{j} \log p_{j}$ is the expected information about $(\Omega , \mathfrak{F}, P)$ gained from observing $X$. Centred around this idea of entropy a mathematical theory of communication was woven by Shannon in a celebrated pair of papers in the 1948 volume of the Bell System Technical Journal. Here Shannon established two fundamental coding theorems about the optimal compressibility of a text in its storage and the optimal capacity of a channel in communicating a text after encoding.

The modern approach to information theory is to view a text in any alphabetic language as a finite time realisation of a stochastic process in discrete time with values in a finite set (called alphabet) and consider the quantity $-\large\frac{1}{n}\normalsize \log P(x_{0}, x_{1}, ..., x_{n-1})$ as the rate at which information is generated by the text $x_{0}, x_{1}, ..., x_{n-1}$ during the period $[0, n-1]$. Under fairly general conditions this rate exhibits an asymptotic stability property as n becomes large. Through the papers of B Mcmillan, A Feinstein, L Breiman, J Wolfowitz and others it is now known that an appeal to this stability property enlarges the scope of Shannon's coding theorems. This gets enriched further by exploiting the Kryloff-Bogoliouboff theory of disintegrating an invariant probability measure into its ergodic components. The first three chapters of this little book are devoted to Shannon's coding theorems and their enriched versions. However, we have not touched upon the coding theorems in their most general form as presented in the book of Te Sun Han.

A decade after the appearance of Shannon's famous work, A N Kolmogorov (1903-1987) demonstrated, rather dramatically, how the notion of the expected rate of generation of entropy or information assumes an intelligence of its own and yields a nonspectral invariant for the classification of dynamical systems. Since very little extra effort is involved in presenting this beautiful work I have taken the liberty of including it as a small digression.

In 1932, while laying the mathematical foundations for quantum mechanics, John von Neumann (1903-1957) introduced the fruitful notion of entropy for the state of a quantum system. If rho is the density operator of the state of a quantum system then its von Neumann entropy S(rho) is defined by S(rho) = Tr rho log rho. Through the work of A S Holevo, B Schumacher, W D Westmoreland and others as outlined in the book of Nielsen and Chuang, the reader can recognise the role of von Neumann entropy in attempts to formulate and establish quantum versions of the coding theorems of Shannon when classical messages are encoded as quantum states and decoding is done by generalised measurements. Our last and the fourth chapter is devoted to a self-contained account of these coding theorems in the quantum avatar as described in the elegant work of A Winter in his 1999 paper.

A large part of the first three chapters of this book does not use anything more than Chebyshev's inequality. The ergodic theorem, martingale theorem and decomposition of an invariant probability measure into its ergodic components are used in arriving at the more sophisticated versions of the classical coding theorems. The last chapter demands nothing more than a knowledge of operators in a finite dimensional Hilbert space.

The present exposition has evolved through the courses of lectures I had given at the Indian Statistical Institute, Calcutta in 1961, the Tata Institute of Fundamental Research, Mumbai in 2001 and 2002, the Institute of Mathematical Sciences, Chennai in 2001 and 2005, the Ramanujan Institute of Advanced Study in Mathematics at the University of Madras in 2005 and Chungbuk National University, Cheongju, Korea in 2005. I am grateful to C R Rao who suggested to me in 1959 the study of information theory for my PhD thesis and J Radhakrishnan, R Parimala, R Balasubramanian, M Krishna, V Arvind, S Parvathi, K Parthasarathy, V Thangaraj and Un Cig Ji who were instrumental in organising these lectures in a congenial atmosphere. I thank Anil Shukla for his elegant TEX of my notes with patience in spite of my repeated requests for changes and corrections. Thanks to the careful proof-reading by P Vanchinathan a significant control over the number of grammatical, typographical and TEX errors has been exercised. The support given by my colleagues at the Delhi Centre of the Indian Statistical Institute is gratefully acknowledged.

13.2. Review by: Kevin Compton.
Mathematical Reviews MR2288798 (2009d:94043).

This book presents a readable exposition of two topics: Shannon's coding theorems and their generalizations concerning asymptotic stability of the information generation rate of a stochastic process; and the quantum versions of these coding theorems where messages are encoded as quantum states and decoded by generalised measurements. The first chapter is a general introduction to codes and Shannon entropy. The second chapter provides the necessary background on ergodicity and martingale convergence before proving the Shannon-Mcmillan-Breiman theorem, which states that the information generation rate of a stationary information source is almost surely convergent. The remainder of the chapter discusses noiseless coding and the Kolmogorov-Sinai entropy of a dynamical system. The third chapter covers ergodic and stationary capacities of noisy channels. The fourth, and final, chapter deals with quantum information theory. Here the von Neumann entropy of a quantum system replaces Shannon entropy in the quantum versions of the coding theorems. The author includes a number of exercises throughout the text.

14.1. From the Preface.

The essential feature of the revised edition is the inclusion of a new chapter devoted to the Knill-Laflamme theory of quantum error correction and its consequences in the construction of t-error correcting quantum codes. Our approach is based on the unification of classical and quantum error correcting codes through imprimitivity systems for finite group actions.

Many typographical error corrections and some minor changes have been made in the text of the first edition.

I have greatly benefited from discussions with V Arvind and Harish Parthasarathy. Ajit Iqbal Singh has rendered valuable help in carefully reading the manuscript and suggesting many improvements. Anil Kumar Shukla has Texed the revised manuscript showing tremendous patience in fulfilling my requests for repeated changes in the text. The continued support of my colleagues in the institute has enabled the completion of this revision in reasonable time. To all of them I express my sincere thanks.