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**Measure and integral. An introduction to real analysis.
2nd ed.**
*(English)*
Zbl 1326.26007

Monographs and Textbooks in Pure and Applied Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-0289-8/hbk). xvii, 514 p. (2015).

The book “Measure and integral” written by Richard L. Wheeden and Antoni Zygmund, treats the Lebesgue integration and related topics, based on the theory of measures, defined and studied in \(n\)-dimensional Euclidean space \(\mathbb{R}^n\).

Preliminaries (Chapter 1) contain results on \(\mathbb{R}^n\), topology, continuous functions and the Riemann integral.

In Chapter 2, the authors introduce the Riemann-Stieltjes integral, preliminary the functions of bounded variation.

The authors define and study the Lebesgue measure of sets in \(\mathbb{R}^n\), in Chapter 3, the base is the outer measure of a set.

In Chapter 4, the authors tackle: Lebesgue measurable functions, semicontinuous functions and the convergence in measure.

Chapter 5 contains the Lebesgue integral, the integral of an arbitrary measure function, the relation between Riemann-Stieltjes and Lebesgue integrals, \(L^p\)-spaces, \(0<p<+\infty\), and Riemann and Lebesgue integrals.

The authors prove the Fubini and Tonelli theorems in Chapter 6, “Repeated integration”.

Chapter 7 treats “Differentiation” and contains the indefinite integral, Lebesgue’s differentiation theorem and the differential in \(\mathbb{R}^n\).

Properties of \(L^p\)-spaces, \(0<p<+\infty\), are contained in Chapter 8, with Young, Hölder, Schwarz and Minkowski inequalities.

The approximations of the identity and maximal functions, Chapter 9, contains convolutions, Young’s theorem, the kernels of Poisson, Fejér and Gauss-Weierstrass, the Hardy-Littlewood maximal functions and the Marcinkiewicz integral.

The abstract integration, Chapter 10, offers additive set functions and measures, Jordan decomposition, measurable functions and integration (Egorov’s theorem), monotone and uniform convergence theorems, Fatou’s lemma, absolutely continuous and singular set functions and measures, Radon-Nikodým theorem, the dual space of \(L^p\), relative differentiation and measures.

Chapter 11, titled “Outer measure”, treats constructing measures from outer measures, metric outer measures, Lebesgue-Stieltjes measure, Hausdorff measure and Carathéodory-Hahn extension theorem.

Chapter 12 contains “A few facts from harmonic analysis” with trigonometric Fourier series.

The Fourier transform, fractional integration and weak derivatives and Poincaré-Sobolev estimates are the subjects of Chapters 13, 14 and 15. Each chapter contains exercises.

Preliminaries (Chapter 1) contain results on \(\mathbb{R}^n\), topology, continuous functions and the Riemann integral.

In Chapter 2, the authors introduce the Riemann-Stieltjes integral, preliminary the functions of bounded variation.

The authors define and study the Lebesgue measure of sets in \(\mathbb{R}^n\), in Chapter 3, the base is the outer measure of a set.

In Chapter 4, the authors tackle: Lebesgue measurable functions, semicontinuous functions and the convergence in measure.

Chapter 5 contains the Lebesgue integral, the integral of an arbitrary measure function, the relation between Riemann-Stieltjes and Lebesgue integrals, \(L^p\)-spaces, \(0<p<+\infty\), and Riemann and Lebesgue integrals.

The authors prove the Fubini and Tonelli theorems in Chapter 6, “Repeated integration”.

Chapter 7 treats “Differentiation” and contains the indefinite integral, Lebesgue’s differentiation theorem and the differential in \(\mathbb{R}^n\).

Properties of \(L^p\)-spaces, \(0<p<+\infty\), are contained in Chapter 8, with Young, Hölder, Schwarz and Minkowski inequalities.

The approximations of the identity and maximal functions, Chapter 9, contains convolutions, Young’s theorem, the kernels of Poisson, Fejér and Gauss-Weierstrass, the Hardy-Littlewood maximal functions and the Marcinkiewicz integral.

The abstract integration, Chapter 10, offers additive set functions and measures, Jordan decomposition, measurable functions and integration (Egorov’s theorem), monotone and uniform convergence theorems, Fatou’s lemma, absolutely continuous and singular set functions and measures, Radon-Nikodým theorem, the dual space of \(L^p\), relative differentiation and measures.

Chapter 11, titled “Outer measure”, treats constructing measures from outer measures, metric outer measures, Lebesgue-Stieltjes measure, Hausdorff measure and Carathéodory-Hahn extension theorem.

Chapter 12 contains “A few facts from harmonic analysis” with trigonometric Fourier series.

The Fourier transform, fractional integration and weak derivatives and Poincaré-Sobolev estimates are the subjects of Chapters 13, 14 and 15. Each chapter contains exercises.

Reviewer: Dan-Mircea Borş (Iaşi)

### MSC:

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

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

01A75 | Collected or selected works; reprintings or translations of classics |

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

28A25 | Integration with respect to measures and other set functions |