Abstract
In the most common construction of the Lebesgue integral of a function, the definition of the integral assumes that one knows the sizes of subsets of the function's domain. In Chapter 1 we introduce measures, the basic tool for dealing with such sizes. The first two sections of the chapter are abstract (but elementary). Section 1.1 looks at sigma-algebras, the collections of sets whose sizes we measure, while Section 1.2 introduces measures themselves. The heart of the chapter is in the following two sections, where we look at some general techniques for constructing measures (Section 1.3) and at the basic properties of Lebesgue measure (Section 1.4). The chapter ends with Sections 1.5 and 1.6, which introduce some additional fundamental techniques for handling measures and sigma-algebras.
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Notes
- 1.
The terms field and σ-field are sometimes used in place of algebra and σ-algebra.
- 2.
See Chap. 8 for some interesting and useful sets that are not Borel sets.
- 3.
If in Example 1.2.1(a) the σ-algebra \(\mathcal{A}\) contains all the subsets of X, then μ is σ-finite if and only if X is at most countably infinite.
- 4.
See items A.12 and A.13 in Appendix A.
- 5.
For details, see Solovay [110].
- 6.
This means that B spans \(\mathbb{R}\) (i.e., that \(\mathbb{R}\) is the smallest linear subspace of \(\mathbb{R}\) that includes B) and that no proper subset of B spans \(\mathbb{R}\). The axiom of choice implies that such a set B exists; see, for example, Lang [80, Section 5 of Chapter III].
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Cohn, D.L. (2013). Measures. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_1
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