A concise introduction to the theory of integration.(English)Zbl 0729.28001

Series in Pure Mathematics, 12. Singapore etc.: World Scientific. vii, 148 p. (1990).
This text covers Lebesgue measure, Lebesgue integration, changes of variables, the Lebesgue spaces, approximate identities and the Radon- Nikodým theorem. Examples and exercises reinforce Lebesgue theory as a synthesis of classical and modern analysis.
The depth of the material is enhanced by the following exercises selected from the text: Exercise (p. 19). Construct $$\psi$$ in C[0,1] such that $$Var(\psi;[0,1])=\infty$$. Exercise (p. 64). Let f be a nonnegative, measurable function on the measure space (E,B,$$\mu$$). If f is integrable, show that $$\lim_{\lambda \to \infty}\lambda \mu (f\geq \lambda)=0$$. Exercise (p. 119). If (E,B,$$\mu$$) is a probability space, show that $$p\in [1,\infty]\to \| f\|_{L^ p(\mu)}$$ is a nondecreasing function for any measurable f on (E,B,$$\mu$$). Exercise (p. 137). Assume that E is a metric space. Show that $$\sigma (C_ b(E;{\mathbb{R}}))$$ coincides with the Borel field $$B_ E$$. Assume, in addition, E is locally compact and show that $$B_ E=\sigma (C_ c(E;{\mathbb{R}})$$. Like these, there are many exercises which give an insight into Lebesgue integration. Proofs of Fatou’s lemma, Fubini’s theorem, the Lebesgue decomposition theorem and the Radon-Nikodým theorem are presented logically and clearly. The get- up and printing are inviting. This text is designed for a semester course at graduate level.

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