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**Lebesgue integration on Euclidean space.**
*(English)*
Zbl 0780.28001

Boston, MA: Jones and Bartlett Publishers. xiv, 588 p. (1993).

This very comprehensive and thoroughly prepared handbook starts with a short chapter presenting basic facts about \(\mathbb{R}^ n\). Then the theory of Lebesgue measure and integration is developed at once in \(\mathbb{R}^ n\). This fact demands a strong effort from the author – for example, the invariance of Lebesgue measure is proved before Fubini’s theorem, but the order of presentation results from his great teaching experience. Thus the notion of abstract sigma algebra appears only in Chapter V, the notion of general measure space – in Chapter VI, and product of abstract measures – in Chapter XI.

The book also includes several examples of application of Lebesgue theory: very long chapters: XIII – on the Fourier transform in several variables, XIV – on Fourier series, XV – on differentiation of integrals in \(\mathbb{R}^ n\) and XVI – on differentiation of functions of one variable. One can find also a chapter on interesting sets (with a “bevy” of Cantor sets) and, rarely included in this kind of books, a chapter dealing with the gamma function. And last, but not least, a collection of about 600 problems for the reader dispersed in the text. Some of them are really non-trivial.

The book also includes several examples of application of Lebesgue theory: very long chapters: XIII – on the Fourier transform in several variables, XIV – on Fourier series, XV – on differentiation of integrals in \(\mathbb{R}^ n\) and XVI – on differentiation of functions of one variable. One can find also a chapter on interesting sets (with a “bevy” of Cantor sets) and, rarely included in this kind of books, a chapter dealing with the gamma function. And last, but not least, a collection of about 600 problems for the reader dispersed in the text. Some of them are really non-trivial.

Reviewer: W.Wilczyński (Łódź)

### MSC:

28-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to measure and integration |

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

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

26B99 | Functions of several variables |