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Unformatted text preview: D Borel and Radon Measures on the Real Line In this appendix we review the theory of signed and complex Borel and Radon measures. These can be defined on locally compact Hausdorff spaces, but for the purposes of this volume we only need to deal with Borel and Radon mea- sures on the real line. This allows several simplifications as compared to the most general theory. We follow the development of abstract measures, Borel measures, and Radon measures given in Folland’s text [Fol99], specializing to the case of measures on the real line. D.1 σ-Algebras It can be shown that the Axiom of Choice implies that there is no way to define a function μ on all of the subsets of R so that all of the following hold: (i) 0 ≤ μ ( E ) ≤ ∞ for every E ⊆ R , (ii) μ ([ a,b ]) = b − a, (iii) if E 1 ,E 2 ,... are finitely or countably many disjoint sets, then μ ( ∪ k E k ) = ∑ k μ ( E k ) , and (iv) μ ( E + h ) = μ ( E ) for all E ⊆ R and h ∈ R . There are several ways to address this. In Appendix B we began with Lebesgue exterior measure | · | e , which does satisfy (i), (ii), and (iv), but fails requirement (iii). Rather unsettlingly, for Lebesgue exterior measure, E ∩ F = ∅ does not imply that | E ∪ F | e = | E | e + | F | e . In order to obtain Lebesgue measure | · | , we therefore dropped require- ment (i), with the result that not all subsets of R are Lebesgue measurable. Although we cannot measure every set, we do have the satisfying fact that requirements (ii), (iii), and (iv) hold for all those sets E that are Lebesgue measurable. On the other hand, there are good reasons for relaxing the requirements in other ways. For example, one of the most important measures is the δ 356 D Borel and Radon Measures on the Real Line measure , which assigns the size 1 or 0 to a set E depending on whether the origin belongs to E or not. Requirements (ii) and (iv) are not satisfied by the δ measure, but both (i) and (iii) do hold. Additional alternatives are to allow a measure to take real or complex values, instead of just nonnegative values. This leads to the idea of signed and complex measures on R . In this appendix we will review the definitions and properties of abstract Borel and Radon measures on the real line. In order to give a useful definition of a measure, we must first decide on the properties that a class of sets should possess in order to be measured. Definition D.1 ( σ-Algebra). A σ-algebra on R is a nonempty collection Σ of subsets of R which satisfies: (a) Σ is closed under both finite and countable unions: E 1 ,E 2 , ··· ∈ Σ = ⇒ uniontext k E k ∈ Σ, (b) Σ is closed under complements: E ∈ Σ = ⇒ E C = R \ E ∈ Σ....
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This note was uploaded on 08/25/2011 for the course MATH 7338 taught by Professor Heil during the Fall '09 term at Georgia Tech.
- Fall '09