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**Theory of the integral. 2nd, revised ed. Engl. translat. by L. C. Young. With two additional notes by Stefan Banach.**
*(English)*
Zbl 0017.30004

Monografie Matematyczne Tom. 7. New York: G. E. Stechert & Co. vi, 347 p. (1937).

This book is an English version of the book which appeared in Polish “Zarys Teorji Calki” (Warszawa) (1930; JFM 57.1392.04) and in French “Théorie de l’intégrale” (Warszawa: Instytut Matematyczny PAN) (1933; Zbl 0007.10501, JFM 59.0266.03).

The present edition differs considerably from the preceding French edition not only because it contains many additional topics treated, but also because the order of material is changed.

Chapters I and II are devoted to the theory of measure and integration of real-valued functions over abstract spaces. Chapter III deals with functions of bounded variation and Lebesgue-Stieltjes integrals over \(n\)-dimensional Euclidean spaces. Chapter IV is devoted to the modern theory of derivation of additive functions of sets and intervals (including derivation in abstract spaces). Considerable attention is given here to the recent work of Ward.

Chapter V deals with the theory of area of a surface \(z = f(x,y)\). In Chapter VI the idea of major and minor functions (“fonctions majorantes et minorantes”) is rather fully exploited; applications to functions of a complex variable (theorems of Looman-Menchoff, and of Besicovich) and to the theory of Perron and of Perron-Stieltjes integrals are particularly noteworthy. Chapter VII deals with functions of generalized bounded variation, important notion which is applied in the next Chapter VIII to the theory of Denjoy integral. An extensive treatment of various problems of the theory of differentiation of functions of two real variables is contained in Chapter IX.

The book closes with two notes of S. Banach, on Haar’s measure and on the Lebesgue integral in abstract spaces. The first of these notes was already in the French edition, while the second has not been published before. There is an extensive bibliography. The short space available for this review does not allow to render justice to a considerable amount of new material and of “fine points” contained in this monograph.

The present edition differs considerably from the preceding French edition not only because it contains many additional topics treated, but also because the order of material is changed.

Chapters I and II are devoted to the theory of measure and integration of real-valued functions over abstract spaces. Chapter III deals with functions of bounded variation and Lebesgue-Stieltjes integrals over \(n\)-dimensional Euclidean spaces. Chapter IV is devoted to the modern theory of derivation of additive functions of sets and intervals (including derivation in abstract spaces). Considerable attention is given here to the recent work of Ward.

Chapter V deals with the theory of area of a surface \(z = f(x,y)\). In Chapter VI the idea of major and minor functions (“fonctions majorantes et minorantes”) is rather fully exploited; applications to functions of a complex variable (theorems of Looman-Menchoff, and of Besicovich) and to the theory of Perron and of Perron-Stieltjes integrals are particularly noteworthy. Chapter VII deals with functions of generalized bounded variation, important notion which is applied in the next Chapter VIII to the theory of Denjoy integral. An extensive treatment of various problems of the theory of differentiation of functions of two real variables is contained in Chapter IX.

The book closes with two notes of S. Banach, on Haar’s measure and on the Lebesgue integral in abstract spaces. The first of these notes was already in the French edition, while the second has not been published before. There is an extensive bibliography. The short space available for this review does not allow to render justice to a considerable amount of new material and of “fine points” contained in this monograph.

Reviewer: J. D. Tamarkin (Providence, R.I.)

### MSC:

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |