The Radon-Nikodym theorem is a fundamental and important result in measure theory and has important applications in modern probability theory. For example, the Radon-Nikodym theorem can be used to prove the existence of conditional expectations (Klenke, 2006). In this chapter, we present some concepts and results related to this theorem.

7.1 Basic Concepts and Facts

Definition 7.1 (Signed Measure). A signed measure on a σ-algebra is a set function μ : ∑ → that is countably additive:

(a) μ() = 0.
(b) For any disjoint sets n ∑, n = 1, 2,…, have


Definition 7.2 (Absolute Continuity). Let (S, ∑, μ) be a measure space and λ a signed measure on ∑. The signed measure λ is said to be absolutely continuous with respect to μ, denoted by λ μ, if and only if μ(A) = 0 (A ∑) implies λ(A) = 0.

Definition 7.3 (Equivalent Measures). Let μ and λ be two measures defined ...

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