×

A user-friendly introduction to Lebesgue measure and integration. (English) Zbl 1339.28001

Student Mathematical Library 78. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2199-1/pbk). ix, 221 p. (2015).
The book under review contains a first-level course in measure theory and integration. It consists of five chapters, which can be divided into three parts. The first part includes Chapter 0 and briefly covers Riemann integration with the approach that is later mimicked in defining the Lebesgue integral, that is, the use of upper and lower sums.
The most remarkable is the second part which contains three chapters. Chapter 1 covers Lebesgue outer measure and Lebesgue measure. Notice that Lebesgue measure is not defined in a standard way – via Carathéodory condition – but in a more “natural” way: a set \(E\subset\mathbb{R}^k\) is Lebesgue measurable if for every \(\varepsilon>0\) there is an open set \(G\) such that \(E\subset G\) and \(m^\ast(G\setminus E)<\varepsilon\). Section 1.3 contains the classic Vitali construction of a non-measurable set. Chapter 2 contains the definition of the Lebesgue integral. It is defined for a bounded measurable function \(f:[a,b]\to\mathbb{R}\), starting from lower and upper sums coming from finite measurable partition of \([a,b]\), similarly to the construction the Riemann integral. The Lebesgue monotone convergence theorem and Fatou’s lemma are obtained as corollaries of Lebesgue dominated convergence theorem. Chapter 3 contains basic informations on \(L^p\) spaces. It is shown that they are Banach spaces. \(L^2\) is shown to be a Hilbert space. At the end of this chapter the author shows that if \(f\in L^2\) then its Fourier series converges to \(f\) in the norm \(|\cdot|_2\).
The last part consists of Chapter 4 and contains a classical abstract measure and integration theory. The author introduces the notions of \(\sigma\)-algebra of sets, measure on \(\sigma\)-algebra, and a measure space. The Carathéodory construction of a measure from an outer measure is presented. The Lebesgue integral is defined via the standard approach which starts from simple functions. Lebesgue convergence theorems and Fatou’s lemma are proved in the usual order. At the last section author considers signed measures and proves the Hahn decomposition theorem. The book includes also an appendix with some suggested projects suitable for end-of-course papers or presentations, like Egorov’s theorem, convergence in measure, Lebesgue’s criterion for Riemann integrability, Fubini’s theorem, etc.
Summing, I fully agree with the opinion posted on the cover of the book: “The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in real analysis.”

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
PDFBibTeX XMLCite