Chapter 20

Representation Theorems20.1 Radon Nikodym Theorem

This chapter is on various representation theorems. The first theorem, the Radon NikodymTheorem, is a representation theorem for one measure in terms of another. The approachgiven here is due to Von Neumann and depends on the Riesz representation theorem forHilbert space, Theorem 19.1.14 on Page 522.

Definition 20.1.1 Let µ and λ be two measures defined on a σ -algebra, S , of subsetsof a set, Ω. λ is absolutely continuous with respect to µ , written as λ ≪ µ, if λ (E) = 0whenever µ(E) = 0.

It is not hard to think of examples which should be like this. For example, supposeone measure is volume and the other is mass. If the volume of something is zero, it isreasonable to expect the mass of it should also be equal to zero. In this case, there is afunction called the density which is integrated over volume to obtain mass. The RadonNikodym theorem is an abstract version of this notion. Essentially, it gives the existence ofthe density function.

Theorem 20.1.2 (Radon Nikodym) Let λ and µ be finite measures defined on a σ -algebra,S , of subsets of Ω. Suppose λ ≪ µ . Then there exists a unique f ∈ L1(Ω,µ) such thatf (x)≥ 0 and

λ (E) =∫

Ef dµ .

If it is not necessarily the case that λ ≪ µ, there are two measures, λ⊥ and λ || such thatλ = λ⊥ + λ ||,λ || ≪ µ and there exists a set of µ measure zero, N such that for all Emeasurable, λ⊥ (E) = λ (E ∩N) = λ⊥ (E ∩N) . In this case the two measures, λ⊥ and λ ||are unique and the representation of λ = λ⊥+λ || is called the Lebesgue decompositionof λ . The measure λ || is the absolutely continuous part of λ and λ⊥ is called the singularpart of λ .

Proof: Let Λ : L2(Ω,µ +λ )→ C be defined by

Λg =∫

Ω

g dλ .

By Holder’s inequality,

|Λg| ≤(∫

Ω

12dλ

)1/2(∫Ω

|g|2 d (λ +µ)

)1/2

= λ (Ω)1/2 ||g||2

where ||g||2 is the L2 norm of g taken with respect to µ +λ . Therefore, since Λ is bounded,it follows from Theorem 17.1.4 on Page 437 that Λ ∈ (L2(Ω,µ + λ ))′, the dual spaceL2(Ω,µ + λ ). By the Riesz representation theorem in Hilbert space, Theorem 19.1.14,there exists a unique h ∈ L2(Ω,µ +λ ) with

Λg =∫

Ω

g dλ =∫

Ω

hgd(µ +λ ). (20.1.1)

597