1.1. Preface to the First Edition of 1933.

The modern theory of real functions became distinct from classical analysis in the second half of the 19th century, as a result of researches, unsystematic at first, which dealt with the foundations of the Differential Calculus or which concerned the discovery of functions whose properties appeared to be very strange and unexpected. The distrust with which this new field of investigation was regarded is typified by the attitude of H Poincaré who wrote, "Autrefois, quand on inventait une fonction nouvelle, c'était en vue de quelque but pratique; aujourd'hui, on les invente tout exprès pour mettre en défaut les raisonnements de nos pères, et on n'en tirera jamais que cela" [in the past new functions were invented with some practical purpose in mind; today they are invented intentionally in order to baffle the reasonings of our fathers, and one cannot deduce anything from them but that].

This view was by no means isolated. Ch Hermite in a letter to T J Stieltjes, expressed himself in even stronger terms: "Je me détourne avec effroi et horreur de cette plaie lamentable des fonctions qui n'ont pas de dérivées".[I turn away with dread and horror from this lamentable plague of functions which have no derivatives] Researches dealing with non-analytic functions and with functions violating laws which one hoped were universal, were regarded almost as the propagation of anarchy and chaos where past generations had sought order and harmony. Even the first attempts to establish a positive theory were rather sceptically received; it was feared that an excessively pedantic exactitude in formulating hypotheses would spoil the elegance of classical methods, and that discussions of details would end by obscuring the main ideas of analysis. It is true that the first researches hardly went beyond the traditional, formal apparatus, fixed by Cauchy and Riemann, which was difficult to adapt to the requirements of the new problems. Nevertheless, these researches succeeded in opening the way to applications of the Theory of Sets to Analysis and - to quote H Lebesgue's inaugural lecture at the College de France - "the great authority of Camille Jordan gave to the new school a valuable encouragement which amply compensated the few reproofs it had to suffer".

R Baire, E Borel, H Lebesgue - these are the names which represent the Theory of real Functions, not merely as an object of researches, but also as a method, names which at the same time recall the leading ideas of the theory. The names of Baire and Borel will be always associated with the method of classification of functions and sets in a transfinite hierarchy by means of certain simple operations to which they are subjected. Excellent accounts of this subject are to be found in the treatises: Ch J de la Vallée. Poussin, Fonctions d'ensemble, Intégrale de Lebesgue, Classes de Baire. 1916, F Hausdorff, Mengenlehre, 1927, H Hahn, Theorie der reellen Funktionen, 1933 (recent edition), C Kuratowski, third volume of the present collection, and finally in the book of W Sierpinski, Topologja ogólna (in Polish), and its English translation, General Topology, to be published in 1934 by the Toronto University Press.

The other line of researches, which arises directly from the study of the foundations of the Integral Calculus, is still more intimately connected with the great trains of thought of Analysis in the last century. On several occasions attempts were made to generalise the old process of integration of Cauchy-Riemann, but it was Lebesgue who first made real progress in this matter. At the same time, Lebesgue's merit is not only to have created a new and more general notion of integral nor even to have established its intimate connection with the theory of measure: the value of his work consists primarily in his theory of derivation which is parallel to that of integration. This enabled his discovery to find many applications in the most widely different branches of analysis and from the point of view of method, made it possible to reunite the two fundamental conceptions of integral, namely that of definite integral and that of primitive, which appeared to be forever separated as soon as integration went outside the domain of continuous functions.

The theory of Lebesgue constitutes the subject of the present volume. While distinguishing it from that of Baire, we have no wish to erect an artificial barrier between two streams of thought which naturally intermingle. On the contrary, we shall have frequent occasion, particularly in the last chapters of this book, to show explicitly how Lebesgue's theory comes to be bound up not only with the results, but also with the very methods, of the theory of Baire. Is not the idea of Denjoy integration at bottom merely a striking adaptation of the idea which guided Baire? Where Baire, by repeated application of passage to the limit, widened the class of functions, Denjoy constructed a transfinite hierarchy of methods of integration starting with that of Lebesgue and whose successive stages are connected by two operations: one corresponding exactly to the generalised integral of Cauchy and the other to the generalised integral of Harnack-Jordan.

Now that the Theory of Real Functions, while losing perhaps a little of the charm of its first youth, has ceased to be a "new" science, it seems superfluous to discuss its importance. It is known that the theory has brought to light regularity and harmony, unhoped for by the older methods, concerning, for instance, the existence of a limit, a derivative, or a tangent. It is enough to mention the theorems, now classical, on the behaviour of a power series on, or near, the boundary of its circle of convergence. Also, many branches of analysis, to cite only Harmonic Analysis, Integral Equations, Functional Operations, have lost none of their elegance where they have been inspired by methods of the Theory of Real Functions. On the contrary, we have learnt to admire in the arguments not only cleverness of calculation, but also the generality which, by an apparent abstraction, often enables us to grasp the real nature of the problem.

The object of the preceding remarks has been to indicate the place occupied by the subject of this volume in the Theory of Real Functions. [Note: In this preface, I made no attempt to write a history of the early days of the theory, and still less to settle questions of priority of discovery, But, since an English Edition of this book is appearing now, I think I ought to mention the name of W H Young, whose work on the theory of integration started at the same period as that of Lebesgue.] Let us now say a few words about the structure of the book. It embodies the greater part of a course of lectures delivered by the author at the University of Warsaw (and published in Polish in a separate book, which has been modified and completed by several chapters. The reader need only be acquainted with a few elementary principles of the Theory of Sets, which are to be found in most courses of lectures on elementary analysis. Actually a summary of the elements of the theory of sets of points is given in one of the opening paragraphs.

Several pages of the book are inspired by suggestions and methods which I owe to the excellent university lectures of my teacher, 'W Sierpinski, the influence of whose ideas has often guided my personal researches. Finally, I wish to express my warmest thanks to all those who have kindly assisted me in my task, particularly to my friend A Zygmund, who undertook to read the manuscript. I thank also Messrs C Kuratowski and H Steinhaus for their kind remarks and bibliographical indications.

S Saks.

Warszaw, May, 1933.

1.2. Preface to the Second Edition of 1937.

This edition differs from the first by the new arrangement of the contents of several chapters, some of which have been completed by more recent results, and by the suppression of a number of errors, obligingly pointed out by Mr V Jarnik, which formed the object of the two pages of Errata in the first edition. It is probable that fresh errors have slipped in owing to modifications of the text, but the reader would certainly find many more, if the author had not received the valuable help of Messrs J Todd, A J Ward and A Zygmund in reading the proofs. Also, Mr L C Young has greatly exceeded his role of translator in his collaboration with the author. To all these I express my warmest thanks.

This volume contains two Notes by S Banach. The first of them, on Haar's measure, is the translation (with a few slight modifications) of the note already contained in the French edition of this book. The second, which concerns the integration in abstract spaces, is published here for the first time and completes the considerations of Chapter I.

The numbers given in the bibliographical references relate to the list of cited works which will be found at the end of the book. The asterisks preceding certain titles indicate the parts of the book which may be omitted on first reading.

S Saks.

Warszaw-Zoliborz, July, 1937.

2.1. Preface (written September 1938).

In the university teaching of analysis one feels, in general, a distinct difference of methods in passing from the real domain (Differential and Integral Calculus) to the complex domain (Theory of Analytic Functions). If in the real domain there is a tendency toward the "arithmetisation" of geometric methods, then in the complex domain the converse process is most frequently applied - the "geometrisation" of analysis. By this "geometrisation" (considering the matter, of course, from a didactical point of view) should be understood not only the use of geometric language and geometric or topological methods, but also the introduction into analysis of certain intuitive geometric concepts, without defining them precisely. Into certain arguments and formulations they enter so strongly that they tend to overshadow purely analytic elements. As an example the classical proof of the Cauchy-Goursat theorem can be given, in which the attention of the beginner is attracted more to the less precise geometric elements of the argument (concerning the curve bounding a region) than to Goursat's basically simple analytical concept.

For didactic as well as other reasons, many authors tried to remove geometric elements from the exposition of the Theory of Functions, applying systematically Weierstrassian methods, which base the definition of a holomorphic function directly on the notion of a power series. Presentations of this type are distinguished by consistency and uniformity; their negative side is the complete renunciation of geometric methods, and because of this, a narrowing of the expanding horizons and perspectives of the Theory itself.

The authors of this book have taken the middle road. By no means renouncing the application of the auxiliary apparatus of Geometry and the Theory of Sets (Topology), they tried to confine it to a domain in which it could be justified and made precise without undue difficulty for the beginner. This domain turned out be sufficient for the proof of such theorems of geometric character as e.g. Runge's theorem (Chapter IV, §§ 1, 2), Riemann's theorem on the mapping of a simply connected region (Chapter V, § 6), the "monodromy" theorem (Chapter IV, § 6), and finally, Cauchy's theorem on the curvilinear integral (Chapter IV, § 2) - in a form not coinciding exactly, to be sure, with the classical formulation, but sufficient for many applications.

This concerns primarily the "elementary" part of the book, i.e. the Introduction and the first six chapters. The last three chapters already have a less elementary character. They make use of strictly analytical methods and embrace more specialised topics: the general theory of entire functions as well as Picard-Landau's theorem (Chapter VII), elliptic as well as modular functions (Chapter VIII), and finally, information about Dirichlet's series and certain fundamental functions such as Euler's "Gamma" and "Beta" functions as well as Riemann's "Zeta" function (Chapter IX).

The division of the book into an "elementary" part and a "special" part requires, by the way, certain reservations. For example, in the first part, the sections denoted by a star can be omitted at the first reading. On the other hand, certain of the topics discussed in the first sections of Chapter VII, as e.g. Weierstrass's theorem on the factorisation of an entire function, or Mittag-Leffler's theorem, obviously belong to the basic knowledge of the domain of the Theory of Functions.

The exercises placed at the ends of the sections have as their aim, primarily, to help the reader to master the methods discussed in the text. The topics of the exercises are in general easy, and the somewhat more difficult ones are supplied with hints. The fact that the exercises are grouped according to methods rather than according to content, and added to the corresponding sections, is in itself helpful to the reader, because it offers him directly the means which he should use in solving them.

In the exercises there were also placed a certain small number of topics which can be considered among the fundamental results of the Theory of Functions, but which do not constitute indispensable links in the structure of the book; we mention e.g. the theorems on sets of the first category of Baire (Introduction, §§ 8 and 11), the elementary proof of the resolution of trigonometric functions into factors ( Chapter I, § 8), the classification of homographic transformations (Chapter I, § 14), certain theorems on power series, such as the theorems of Fejér, Tauber (Chapter III, § 2), Jentzsch (Chapter IV, § 3), Fatou, and M Riesz (Chapter VI § 3), Hadamard's "three circle" theorem (Chapter III, § 12), Blaschke's theorem on the roots of a bounded function (Chapter IV, § 4 and Chapter VII, § 2), classical proofs of the theorems of Picard and Montel, based on properties of modular functions (Chapter VII, § 2), the inequalities of Carathéodory (Chapter VII, § 10), and quasi-normal families of functions (Chapter VII, § 13).

The entire content of the book is theoretically accessible to the first year student, for it presupposes only a knowledge of the arithmetic of complex numbers and certain information concerning the convergence of sequences and series, the continuity of functions, etc., - information included now-a-days even in the programs of lyceums. Other helpful information (from the Theory of Sets and Topology) has been given in the Introduction, and partly also (from Analysis) in Chapter I.

Factually, however, the book demands from the reader a certain familiarity with the methods of abstract thinking. This concerns, first of all, the Introduction. The beginner may initially limit himself to only a cursory examination of the Introduction, in order to orient himself in the terminology, and acquaint himself better with it as he reads the succeeding chapters.

Many persons have given us help in editing this book. Miss Stefania Braun, with unusual conscientiousness, collaborated with us in the proof reading. We are indebted to her for the elimination of many errors and oversights - not only misprints. Mr Bronislaw Knaster gave us valuable advice concerning make-up. Mr Edward Otto was kind enough to draw the figures. To all of them we sincerely express our thanks.

September 1938 .
S Saks, A Zygmund.

2.2. Preface to the English edition of 1952.

Stanislaw Saks was a man of moral as well as physical courage, of rare intelligence and wit. To his colleagues and pupils he was an inspiration not only as a mathematician but as a human being. In the period between the two world wars he exerted great influence upon a whole generation of Polish mathematicians in Warsaw and Lwów. In November 1942, at the age of 45, Saks died in a Warsaw prison, victim of a policy of extermination.

The present book owes him much more than the mere fact of co-authorship would indicate. In particular, the general idea of the approach to the theory was his. For this reason it seemed desirable to preserve the character of the presentation. Only minor changes and indispensable corrections have been introduced in the English edition.

In the reading of the proofs of this edition help was given by Prof E J Scott, translator of the book, and by Mr J Panz, J Feldman and L Gordon. Especially valuable was the help of Mr Gordon, who went very thoroughly through the text and corrected a number of inaccuracies. Professor S Eilenberg corrected a slip in the proof of a theorem. To all these persons the undersigned wishes to express his sincere gratitude.

Chicago, October 1952
A Zygmund