Extension of Measures

In Chapter 2 (see Example 2C) we considered the Lebesgue measure $$\lambda$$ on the Borel algebra $$\mathfrak{R}$$ , which is the $$\sigma$$ -algebra of subsets of the real line $$\mathbb{R}$$ generated by the collection of all open intervals; and we have been using the notion of Lebesgue measure since then, although it has not been properly constructed so far. Indeed, in Example 2C we promised to prove existence and uniqueness of the Lebesgue measure $$\lambda: \mathfrak{R}\rightarrow \overline{\mathbb{R}}$$ in Chapter 8 . We will comply with that promise in Section 8.3, as a special case of the following program. (1) First we introduce the concept of a measure $$\mu$$ on an algebra $$\mathcal{A}$$ (rather than on a $$\sigma$$ -algebra) of subsets of set X. (2) Then we consider the notion of an outer measure $$\mu ^{{\ast}}$$ generated by that measure $$\mu$$ on an algebra $$\mathcal{A}$$ , which is a set function on the power set $$\wp (X)$$ . (3) Finally, we show that this outer measure $$\mu ^{{\ast}}$$ induces a $$\sigma$$ -algebra $$\mathcal{A}^{{\ast}}$$ of subsets of X (such that $$\mathcal{A}\subseteq \mathcal{A}^{{\ast}}$$ ) upon which the restriction $$\mu ^{{\ast}}\vert _{\mathcal{A}^{{\ast}}}$$ is a measure on the $$\sigma$$ -algebra $$\mathcal{A}^{{\ast}}$$ . This is the Carathéodory Extension Theorem, which is the central result of this chapter, whose applications go as far as Chapters 9 , 11 , and 13 .