Measure and Integration Theory (de Gruyter Studies in Mathematics)

This is a focused and completely self-contained book on measure spaces and integration. The only topics I think the book would be improved by including are absolutely continuous functions and functions of bounded variation.Chapter I starts with systems of subsets of a fixed set. Bauer gives a clean theory of rings of sets, algebras of sets, sigma-algebras of sets, and Dynkin systems (lambda-systems). Indeed we usually want to talk about sigma-algebras, but developing a theory of these other objects makes theorems and techniques systematic rather than ad hoc. After working out the theory of the above systems of sets, Bauer introduces contents and premeasures on rings of sets, outer measures, and the sigma-algebra of sets that are measurable with respect to an outer measure (the Carathéodory extension theorem).In Chapter II, Bauer cleanly and precisely defines what it means for a function to be integrable. He is precise about when he is talking about functions taking values in the extended real numbers, nonnegative extended real numbers, and real numbers. Happily, Bauer talks about the seminormed space of integrable functions rather than the Banach space of equivalence classes of integrable functions. Indeed one usually wants to speak about the Banach space, but talking about elements of this Banach space rather than actual functions hides some of what is going on, and makes it impossible to say whether one is talking about a particular function or an equivalence class of functions. The various convergence theorems (Fatou's lemma, monotone convergence theorem, dominated convergence theorem) are all proved. The Radon-Nikodym theorem and the change of variables theorem are also proved in Chapter II, and the notion of an equi-integrable collection of functions is defined.Chapter III is on products of sigma-finite measures. It gives good proofs of Tonelli's theorem and Fubini's theorem, and makes clear the different ways in which they are used rather than having them bleed together into a naive idea of being able to manipulate product/iterated integrals any way that seems to make sense.Finally, Chapter IV is about measures on topological spaces. Bauer defines a Borel measure as a measure on the Borel sigma-algebra of a Hausdorff space that assigns finite values to compact sets. He defines the notion of locally finite, inner regular, and outer regular measures, and defines a Radon measure as a locally finite and inner regular measure on the Borel sigma-algebra of a Hausdorff space; in particular, a Radon measure is a Borel measure. He proves for example that for a first-countable Hausdorff space, an inner regular Borel measure is a Radon measure. Much of this chapter is results about Borel measures on Polish spaces, and Bauer defines the notions of vague and weak convergence of measures and proves the portmanteau theorem.

I got the product as promised without any problems.