Extension of Measures

Carlos S. Kubrusly
Additional contact information
Carlos S. Kubrusly: Catholic University of Rio de Janeiro, Electrical Engineering Department

Chapter 8 in Essentials of Measure Theory, 2015, pp 131-157 from Springer

Abstract: Abstract 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 .

Keywords: Outer Measure Zero; Vitali Set; Infinite Countable Family; Finite Additivity; Pairwise Disjoint Sets (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-22506-7_8

Ordering information: This item can be ordered from
http://www.springer.com/9783319225067

DOI: 10.1007/978-3-319-22506-7_8

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().