This book is addressed mainly to Ph.D. students who have graduated from distinct universities where they have been introduced to general topology, measure theory, integrals and probability theory but their knowledge may not be unified and the students may not see well significant applications of topology to measure theory nor the influence of measure theory on research in topology. The author aims to recall basic facts known from courses of topology and measure theory, show some fundamental constructions in topology and measure theory, with special emphasis on the interplay between several important topics of measure theory, topology and category theory. This book can be recommended also to university teachers of mathematics and researchers who need to apply topology and measure theory.
The book is divided into an Introduction and the following three chapters: Chapter 1 (Spaces), Chapter 2 (Towards the notion of a product), Chapter 3 (Measure – or integral?). The book ends with the following: A shortlist of symbols and abbreviations, Index and Bibliography. Unfortunately, it is left to the readers to establish what the set-theoretic framework for this book is. The author uses freely the axiom of choice and category theory.
Chapter 1 consists of Sections 1.1–1.6. In Section 1.1, the author recalls some basic definitions and facts concerning topological and metric spaces, shows proofs in ZFC of Urysohn’s lemma and the Tietze-Urysohn Extension Theorem, emphasizes the role of metric spaces in topology. In Section 1.2, the author recalls the notions of a \(\sigma\)-field, \(\sigma\)-ring, measurable space and a measurable function; furthermore, among other notions relevant to measurable spaces, the author introduces the notion of a \(\delta\)-ring to apply it to constructions of \(\sigma\)-fields and \(\sigma\)-rings generated by a collection of subsets of a given set. Section 1.3 is devoted to the abstract notion of a measure in a measurable space and construction of the Lebesgue measure on the real line. Section 1.4 concerns measures on \(\sigma\)-fields and \(\sigma\)-rings, in particular, Borel measures in metric spaces. Section 1.5 concerns outer measures and measurable sets in the sense of Carathéodory. In Section 1.6, the author recalls the notions of a compact set in a topological space, a locally compact space, algebras of continuous real functions separating points, the Stone-Weierstrass and Gelfand-Kolmogorov theorems. Compact and locally compact spaces are assumed to be Hausdorff. Moreover, in Section 1.6, the author discusses the Alexandorff compactification of a non-compact locally compact space to show some important properties of locally compact spaces, their algebras of continuous real functions, and Baire sets.
Chapter 2 is divided into the following sections: Section 2.1 (Categories), Section 2.2 (Topologies and \(\sigma\)-fields), Section 2.3 (Products of measures), Section 2.4 (Subspaces of products). The author uses category theory to discuss products of topological spaces, products of \(\sigma\)-rings and products of measures. It may be hard for a reader not familiar with category theory to understand the categorical approach to products. The Lebesgue measure in \(\mathbb{R}^n\) is discussed in Section 2.3. An introduction to compactifications in ZFC is given in Section 2.4.
The unique section of Chapter 3 is titled “The Lebesgue integral”. It is divided into six subsections. It contains an axiomatic approach to the Lebesgue integral, basic properties of the Lebesgue integral with respect to a given measure, basic facts about \(L^p\) spaces.