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**A concise introduction to the theory of integration.
2nd ed.**
*(English)*
Zbl 0826.28001

Basel: Birkhäuser. viii, 184 p. (1994).

This is the second edition of a text introducing Lebesgue measure and integration theory in Euclidean spaces together with important aspects of traditional advanced calculus and some abstract measure theory, intended for a broad audience of graduate students studying mathematics and other mathematically-related areas. The second edition is a bit longer than the first one (1990) due to the addition of two new sections, solutions for about one-third of the problems, and a list of symbols. The appearance is crisper and some of the exposition (e.g., the proof of the Daniell-Stone representation theorem) has been modified, but in most regards reviews of the first edition (M.R. 93e:28001; Zbl 0729.28001) still apply.

The concise nature of the text arises from brevity to some extent, but as much from the efficiency and sophistication of the approach. For example, Lebesgue measure is developed in \(\mathbb{R}^n\) from the start, using \(\lambda\) and \(\pi\) systems, and the Hardy-Littlewood maximal inequality is used to obtain the Lebesgue differentiation theorem. While the preface states that “nothing has been done in ‘complete generality’ ”, the coverage is quite extensive, including, for example, Lieb’s version of Fatou’s lemma and amalgams of \(L^p\) spaces. The coverage of “advanced calculus” topics includes development of surface measure as the derivative of Lebesgue measure across a surface, change of variables and the divergence theorem in general Euclidean spaces. The abstract theory includes extensions of measures, Lebesgue spaces, convolutions, and the Radon-Nikodým theorem. There are a number of valuable applications and a good collection of problems, many of which introduce substantial applications or investigate deeper aspects of the theory. The text suffers from the lack of a bibliography and gains some efficiency through assuming a good bit of knowledge on the part of the reader without much in the way of reminders but, taken all together, it is a very interesting, well-informed book which draws on recent approaches not found in several commonly used texts.

The concise nature of the text arises from brevity to some extent, but as much from the efficiency and sophistication of the approach. For example, Lebesgue measure is developed in \(\mathbb{R}^n\) from the start, using \(\lambda\) and \(\pi\) systems, and the Hardy-Littlewood maximal inequality is used to obtain the Lebesgue differentiation theorem. While the preface states that “nothing has been done in ‘complete generality’ ”, the coverage is quite extensive, including, for example, Lieb’s version of Fatou’s lemma and amalgams of \(L^p\) spaces. The coverage of “advanced calculus” topics includes development of surface measure as the derivative of Lebesgue measure across a surface, change of variables and the divergence theorem in general Euclidean spaces. The abstract theory includes extensions of measures, Lebesgue spaces, convolutions, and the Radon-Nikodým theorem. There are a number of valuable applications and a good collection of problems, many of which introduce substantial applications or investigate deeper aspects of the theory. The text suffers from the lack of a bibliography and gains some efficiency through assuming a good bit of knowledge on the part of the reader without much in the way of reminders but, taken all together, it is a very interesting, well-informed book which draws on recent approaches not found in several commonly used texts.

Reviewer: J.W.Hagood (Oxford/Ohio)

### MSC:

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

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |

26B15 | Integration of real functions of several variables: length, area, volume |