1 - Measure-theoretic preliminaries

Summary

1. Discussion. In this opening chapter, we offer a review of the basic facts we need from measure theory for the rest of the book (it doubles as an introduction to our pedagogic method). For readers seeking a true introduction to the subject, we recommend first perusing, e.g. Folland (1984); experts meanwhile may safely jump to Chapter 2.

2. Comment. When an exercise is given in the middle of a proof, the end of the exercise will be signaled by a dot:

The conclusion of a proof is signaled by the box sign at the right margin, thus:

Basic definitions

In this subchapter, we discuss algebras, σ-algebras, generation of a σ-algebra by a family of subsets, completion with respect to a measure and relevant definitions.

3. Definition. Let Ω be a set. An algebra of subsets of Ω is a non-empty collection A of subsets of Ω that is closed under finite unions and complementation. A σ-algebra is a collection A of subsets of Ω that is closed under countable unions and complementation.

4. Comment. Every algebra of subsets of Ω contains the trivial algebra {∅, Ω}.

5. Exercise. Let Ω be a set and let C be a family of subsets of Ω. Show that the intersection of all σ-algebras of subsets of Ω containing C is itself a σ-algebra.