1.1. Review by: J D Tamarkin.
Bull. Amer. Math. Soc. 40 (1) (1934), 16-18.

The present volume ... is a translation, entirely revised and augmented by several important chapters, of the author's Polish book, Zarys Teorji Calki, Warsaw, 1930. It fills in a serious gap in the literature of the real function theory, and of the theory of differentiation and integration, which has been acutely felt during all the recent period of vigorous growth and development of these disciplines. While Hermite, together with a large part of his contemporaries and immediate successors, including Poincaré, contemplated with horror the pathological cases of functions without derivatives, the ideas and methods created precisely for handling such bad cases turned out to be extremely fruitful and indispensable for the treatment of most classical and venerated problems of analysis. Far from bringing in the feared anarchy and disorder, they have allowed us in many instances to reach a harmony and completeness of results which were entirely out of the reach of older classical methods. The problems which require essentially a fundamental knowledge of the most modern chapters of the real function theory are "everywhere dense" in the field of analysis, and the mathematician who wants to make any headway in the theory of functions of a complex variable, of differential and integral equations, of Fourier series, and the like, will find himself badly handicapped if he is not well equipped on the side of the real function theory, and particularly of the theory of differentiation and integration. But these are precisely the chapters which so far have been lacking an adequate treatment in the literature. Except for the treatise of Hobson, which still holds its honourable place, we are unable to name any other book or monograph where a serious attempt has been made to unify the numberless results scattered throughout all mathematical periodicals and to give a general picture of the present state of the subject. Such an attempt, and a fully successful one, is represented by the book of Saks. Its main characteristic unifying feature is the systematic use, from the very beginning, of the notion of additive functions of sets. This allows him to bring in, with greatest clearness and economy of thought and notation, the descriptive theory of various integrals he discusses. At the same time it leads very naturally to fundamental problems of the modern differentiation theory, and to an extensive use of Dini's derived numbers and of approximate derivatives and derived numbers, as well as to the question of the unique determination of an additive function of sets by means of its derived numbers of various sorts. The systematic development of these notions constitutes the very nerve of the book. A mathematician who will not avail himself of the rich source of information brilliantly presented in Saks' book will deprive himself of the use of valuable ideas and tools which would be of help to him during all his "analytical" career. In publishing the present monograph and the preceding one by Banach, the Monografje Matematyczne have set up a high standard which will be difficult to match. We congratulate them on this happy start and we wish to attract to this important undertaking all attention of the mathematical world, which it justly deserves. ...

In conclusion we may state our conviction that Saks's book is eminently suited to be used in a graduate course on the real function theory. This opinion has been checked by the reviewer's personal experience.

1.2. Review by: A S Besicovitch.
The Mathematical Gazette 19 (232) (1935), 57-58.

The book starts with the theory of functions of an elementary figure. The main results of the theory are established, which allows the author to escape repeating the same argument at various occasions and which also brings unification in the methods used in the book.

Then follows a very thorough and complete development of the theory of the Lebesgue measure and of the Lebesgue, Denjoy and Perron integration, and of the mutual relation of these processes of integration (the Hake and Alexandroff-Looman theorems are given).

Apart from the pure theory of integration the following problems are considered:

(1) Area of a surface $z = f(x, y)$. The author goes as far as to give the Tonelli theorem.

(2) Properties of functions of a single variable with respect to differentiation. The main Denjoy results are fully represented.

(3) Differentiability of functions of two variables. The author gives an account of work of H Rademacher, W Stepanoff and U S Haslam-Jones.

The book is concluded by the chapter on Lebesgue integrals in abstract spaces and by a note by S Banach on the Haar measure.

The book as a whole is planned remarkably well and also one feels that all the details have been thought over very carefully. The whole subject of the book is systematically developed on deep principles of the modern Analysis. From the point of view of the material represented in the book it is valuable both for an expert in the subject and for a beginner. For all that, the author has succeeded in making the book quite easy to read.

The book is a great event in the literature on modern Analysis and an excellent success for the collection to which it belongs (Mathematical Monographs, Warsaw).

2.1. Review by: J D Tamarkin.
Bull. Amer. Math. Soc. 44 (9.1) (1938), 615-616.

This is the third edition of the excellent and eminently useful book by Saks (the first appeared in 1930, in Polish, and the second in 1933, in French; the latter was reviewed in this Bulletin. It is, however, almost a new book, due to numerous changes in exposition and order of the material and important additions of new topics treated. The opening chapter, I (The integral in an abstract space), treats of the modern theory of abstract measure and integration. The basis is specialised in Chapter II (Carathéodory measure) and Chapter III (Functions of bounded variation and the Lebesgué-Stieltjes integral). Chapter IV (Derivation of additive functions of a set and of an interval) contains a considerable amount of new material, in particular an exposition of important investigations of Ward. It is followed by Chapter V (Area of a surface $z = f(x, y)$) and Chapter VI (Major and minor functions) which contains an elegant treatment of the Perron integral and applications to the theory of functions of a complex variable (Looman-Menchoff theorem). Results of Chapter VII (Functions of generalised bounded variation) are used in the subsequent Chapter VIII (Denjoy integrals). The last chapter, IX (Derivates of functions of one or two real variables), contains a thorough exposition of results of Banach, Besicovitch, Denjoy, Khintchine, Ward, and many other authors. The book closes with two appendices by Banach (On Haar's measure, and The Lebesgue integral in abstract spaces) and with a ten page list of references. The excellent qualities of the book were sufficiently pointed out in the review of the French edition; they explain the remarkable success of the book. The reviewer has no doubt that a fourth edition, still further improved and augmented, will appear before long.

2.2. Review by: Børge Jessen.
Matematisk Tidsskrift. B, tematisk tidsskrift. B (1938), 65-67.

The first edition of Saks' Book, written in French, was published in 1933, but was sold out in 1935. It is gratifying that this important book is now available. At the same time as the translation into English, made by L C Young, the author has used the opportunity for a repositioning of some of the chapters, whereby the abstract point of view is brought to the foreground, and for an extension various recent studies. ... the book differs greatly from the classic books of Lebesgue and Carathéodory, being more limited in the scope of the material treated to give a more rounded production. ... for anyone who wants to know the theory in its present scope, the book is a great aid; of particular value in this connection are the detailed references to the literature.

2.3. Review by: John Todd.
The Mathematical Gazette 22 (248) (1938), 84-85.

For a book of this type to achieve what is essentially a third edition within seven years is unusual; its outstanding success must be very gratifying not only to Dr Saks and the Editorial Committee of the Monografie Matematyczne, but also to analysts everywhere in so far as it is evidence of the growing interest in our subject. In these circumstances it will be all but sufficient to describe some of the considerable changes, both in material and in arrangement, which have been made in the French edition reviewed in the Gazette, XIX, p. 17 and which has been, appropriately, translated by L C Young.

The general impression produced by the new edition is that it is more of a monograph and less of a textbook; on the other hand, the book is not encyclopaedic, and all will appreciate the nice judgment exercised by Dr Saks in his choice of material and his methods of presentation of it and, in particular, in his avoidance of generalisations of uncertain importance. Dr Saks' book will be invaluable to the advanced student and the active researcher, while the novice will find it very stimulating and he will be encouraged to continue his studies of this fundamental part of the Modern Theory of Real Functions.

3.1. From the Publisher.

An excellent introduction to modern real variable theorem, this volume covers all the standard topics: theory, theory of measure, functions with general properties, and theory of integration, with emphasis on the Lebesgue integral and its related theory of derivation.

The author begins with a discussion of the integral in an abstract space, covering additive classes of sets, measurable functions, integration of sequences of functions, and the Lebesgue decomposition of an additive function. Succeeding chapters cover Carathéodory measure; functions of bounded variation and the Lebesgue-Stieltjes integral; the derivation of additive functions of a set and of an interval; and major and minor functions and the Perron integral. Additional topics include functions of generalised bounded variation; Denjoy integrals; and derivates of functions of one or two real variables.

This book will prove to be extremely useful as a course text or as supplementary reading to students of real variable theory and others interested in this branch of mathematics. Only a minimal background in elementary analysis is necessary, and the preface offers a helpful overview of the history of the theory of real functions.

4.1. Review by: Maurice Heins.
Bull. Amer. Math. Soc. 60 (5) (1954), 495-497.

The theory of analytic functions has figured as a standard topic in the curriculum of a mathematics student for many years and there is a fairly clearcut agreement on what makes up the minimal contents of a first course on the theory of functions of a complex variable. However, when one examines the large number of texts on the subject, it is evident that standards of rigour and generality vary considerably. It is also clear that one can discern two quite distinct threads running through the fabric: first, the presence of arguments and methods which are very general - such as the use of topological notions (connectedness, compactness, interiority, etc.) - and are not peculiar to the theory of analytic functions; second, the presence of results and methods which are due to the particular features of the theory of analytic functions and give the theory its special colour. In exposing a classical discipline, it is highly desirable that one should seek to isolate on the one hand the general tools and methods which are of constant use but are not peculiar to the discipline and on the other hand those features of the discipline which are special to it.

Such a program is envisaged by the book under review, the Analytic functions of Saks and Zygmund translated into English by E J Scott. Ever since its appearance in 1938, the original Funkcje Analityczne has evoked considerable interest far beyond the Polish mathematical public; tantalising reference was frequently made to the manner in which certain topics were treated, a notable example being the elementary treatment of the Runge theorem concerning the approximation of analytic functions by polynomials and the exploitation of this result to prove the Cauchy integral theorem for simply-connected regions of the finite plane (i.e. regions having connected complement with respect to the extended plane) and to pave the way for the treatment of the Riemann mapping theorem. ...

This book is a very welcome addition to the collection of texts on the theory of analytic functions which are now available in English. It will be a rewarding experience to the earnest student.

4.2. Review by: E C Zeeman.
The Mathematical Gazette 39 (327) (1955), 79-80.

The book, Funkcje Analityczne, was written in 1938. This translation by E J Scott is the first English edition, and is indeed most welcome. It is a thorough and comprehensive text, covering considerably more than most introductions to the theory; but its chief assets are its clarity and originality of presentation. It is roughly divided into two: the first six chapters leading up to conformal transformations and analytic functions, while the last three chapters are on specialised functions.

In the first half, as the authors point out in the preface, they steer the middle road between the "geometric" and "arithmetic" approaches to the subject, in that they make full use of topological ideas, but take nothing on trust. All the necessary foundations are proved and intuitive geometric notions are defined precisely, which is most satisfying to the student. The book therefore starts with a good introduction of forty pages on topology. The Jordan Curve Theorem is a typical example of a result which is assumed in nearly all other expositions, but which the authors here avoid, since they obviously have not space to prove it. It is however proved for a polygon, for use in the Schwarz-Christoffel formulae. At first sight this seems a serious limitation, say for Cauchy's Theorem, but a reasonably general case of the latter is achieved by early introduction of Runge's Theorem (expressing a holomorphic function as the limit of polynomials).

The net result is a sophisticated approach, which would be fairly hard going for a beginner, but which reaps elegant rewards later (such as Riemann's Theorem on conformal mapping and Picard's Theorems). For instance the student is asked to master normal families of functions (i.e. families in which every sequence has an almost uniformly convergent subsequence) before he is allowed to see the exponential function. So that although the preface claims "the entire content is theoretically accessible to the first year student", it is more suited to the second and third years. On the other hand, the style is delightfully lucid, particularly at the very points which cause students most trouble, such as the complex number sphere, the branches of a function, and singularities. The chapter explaining analytic functions themselves (by the Weierstrassian approach) is a very full and beautiful treatment, the best the reviewer has seen.

Summing up: this is the book for the brighter student. It is written with an eye to the "expanding horizons and perspectives of the Theory", and provides an invaluable introduction, not only to the deeper complex variable theory, but to differential geometry

Review by: E M Wright.
The Mathematical Gazette 54 (388) (1970), 187-188.

The Polish edition of this book appeared in 1938 and an English translation in 1952. This (second) English edition differs from the first by the addition of a chapter on harmonic and subharmonic functions by Professor Zygmund, the surviving author. The book begins with an Introduction on the Theory of Sets and on this foundation is built an elaborate and far-reaching edifice of complex variable theory. The topics dealt with include holomorphic and meromorphic functions, geometric methods, conformal transformations, analytic functions, entire functions (including Picard's theorem), elliptic functions, the Gamma and Zeta functions and Dirichlet series. The topics are as soundly chosen and elegantly and clearly expounded as one would expect from the two authors. The book is the best sort of standard text-book, one that is worth keeping for reference purposes long after one has read it.