proof of Radon-Nikodym theorem

The following proof of Radon-Nikodym theorem is based on the original argument by John von Neumann. We suppose that μ and ν are real, nonnegative, and finite. The extension to the σ-finite case is a standard exercise, as is μ-a.e. uniqueness of Radon-Nikodym derivativeMathworldPlanetmath. Having done this, the thesis also holds for signed and complex-valued measuresMathworldPlanetmath.

Let (X,) be a measurable spaceMathworldPlanetmathPlanetmath and let μ,ν:[0,R] two finite measures on X such that ν(A)=0 for every A such that μ(A)=0. Then σ=μ+ν is a finite measure on X such that σ(A)=0 if and only if μ(A)=0.

Consider the linear functional T:L2(X,,σ) defined by

Tu=Xu𝑑μuL2(X,,σ). (1)

T is well-defined because μ is finite and dominated by σ, so that L2(X,,σ)L2(X,,μ)L1(X,,μ); it is also linear and boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath because |Tu|uL2(X,,σ)σ(X). By Riesz representation theoremMathworldPlanetmath, there exists gL2(X,,σ) such that

Tu=Xu𝑑μ=Xug𝑑σ (2)

for every uL2(X,,σ). Then μ(A)=Ag𝑑σ for every A, so that 0<g1 μ- and σ-a.e. (Consider the former with A={xg(x)0} or A={xg(x)>1}.) Moreover, the second equality in (LABEL:eq:q) holds when u=χA for A, thus also when u is a simple measurable functionMathworldPlanetmath by linearity of integral, and finally when u is a (μ- and σ-a.e.) nonnegative -measurable function because of the monotone convergence theoremMathworldPlanetmath.

Now, 1/g is -measurable and nonnegative μ- and σ-a.e.; moreover, 1gg=1 σ- and μ-a.e. Thus, for every A,

A1g𝑑μ=A𝑑σ=σ(A) (3)

Since σ is finite, 1/gL1(X,,μ), and so is f=1g-1. Then for every A

Title proof of Radon-Nikodym theorem
Canonical name ProofOfRadonNikodymTheorem
Date of creation 2013-03-22 18:58:03
Last modified on 2013-03-22 18:58:03
Owner Ziosilvio (18733)
Last modified by Ziosilvio (18733)
Numerical id 5
Author Ziosilvio (18733)
Entry type Proof
Classification msc 28A15
Synonym Hilbert spacesMathworldPlanetmath proof of Radon-Nikodym’s theorem
Synonym measure- theoretic proof of Radon-Nikodym theorem