# CHAPTER 7

# THE RADON-NIKODYM THEOREM

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:

_{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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