In this series of appendices we recall several basic concepts and facts in measure theory, topology and functional analysis that are useful throughout the book. Our purpose is to provide the reader with a quick, accessible source of references to measure and integration, general and differential topology and spectral theory, to try and make this book as self-contained as possible. We have not attempted to make the material in these appendices completely sequential: it may happen that a notion mentioned in one section is defined or discussed in more depth in a later one (check the index).

As a general rule, we omit the proofs. For Appendices A.1, A.2 and A.5, the reader may find detailed information in the books of Castro [Cas04], Fernandez [Fer02], Halmos [Hal50], Royden [Roy63] and Rudin [Rud87]. The presentation in Appendix A.3 is a bit more complete, including the proofs of most results, but the reader may find additional relevant material in the books of Billingsley [Bil68, Bil71]. We recommend the books of Hirsch [Hir94] and do Carmo [dC79] to all those interested in going further into the topics in Appendix A.4. For more information on the subjects of Appendices A.6 and A.7, including proofs of the results quoted here, check the book of Halmos [Hal51] and the treatise of Dunford and Schwarz [DS57, DS63], especially Section IV.4 of the first volume and the initial sections of the second volume.

Measure spaces

Measure spaces are the natural environment for the definition of the Lebesgue integral, which is the main topic to be presented in Appendix A.2.We begin by introducing the notions of algebra and σ-algebra of subsets of a set, which lead to the concept of measurable space. Next, we present the notion of measure on a σ-algebra and we analyze some of its properties. In particular, we mention a few results on the construction of measures, including Lebesgue measures in Euclidean spaces. The last part is dedicated to measurable maps, which are the maps that preserve the structure of measurable spaces.

Measurable spaces

Given a set X, we often denote by Ac the complement X \ A of each subset A.