martingale proof of the Radon-Nikodym theorem

We apply the martingale convergence theorem to prove the Radon-Nikodym theoremMathworldPlanetmath, which states that if μ and ν are σ-finite measuresMathworldPlanetmath on a measurable spaceMathworldPlanetmathPlanetmath (Ω,) and ν is absolutely continuousMathworldPlanetmath with respect to μ then there exists a non-negative and measurable f:Ω such that ν(A)=Af𝑑μ for all measurable sets A.

As any σ-finite measure is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to a probability measure (, it is enough to prove the result in the case where μ and ν are probability measures. Furthermore, by the Jordan decomposition, the result generalizes to the case where ν is a signed measure. So, we just need to prove the following.

Theorem (Radon-Nikodym).

Let P and Q be probability measures on the measurable space (Ω,F), such that Q is absolutely continuous with respect to P. Then, there exists a non-negative random variableMathworldPlanetmath X such that EP[X]=1 and Q(A)=EP[1AX] for every AF.

Here, X is called the Radon-Nikodym derivative of with respect to .

More generally, for any sub-σ-algebra 𝒢 of we can restrict the measures and to 𝒢 and ask if the Radon-Nikodym derivative of |𝒢 with respect to |𝒢 exists. If it does we shall denote it by X𝒢, which by definition is a non-negative 𝒢-measurable random variable satisfying (A)=𝔼[1AX𝒢] for all A𝒢.

We note that if X𝒢 does exist, then it is uniquely defined (-almost everywhere). Suppose that X~𝒢 also satisfied the required properties, then


so X𝒢X~𝒢 almost surely. Similarly, X~𝒢X𝒢 and therefore X~𝒢=X𝒢 (almost surely).

First, the easy case. For a finite σ-algebra, the Radon-Nikodym derivative can be written out explicitly.

Lemma 1.

If G is a finite sub-σ-algebra of F then the Radon-Nikodym derivative XG exists.


Let A1,A2,,An be the minimalPlanetmathPlanetmath non-empty elements of 𝒢. These are pairwise disjoint subsets of Ω such that every set in 𝒢 is a union of a subcollection of the Ak. Set


Note that whenever (Ak)=0 then (Ak)=0, and we adopt the convention that 00=0. Clearly, X𝒢 is 𝒢-measurable, and


Here, we have used 𝔼[1Ak]=(Ak) and 1AkAj=0. By linearity, this equality remains true if both sides are replaced by any union of the Ak, and therefore X𝒢 is the required Radon-Nikodym derivative. ∎

Next, martingaleMathworldPlanetmath convergence is used to prove the existence of the Radon-Nikodym derivative in the case where the σ-algebra 𝒢 is separablePlanetmathPlanetmath. By separable, we mean that there is a countableMathworldPlanetmath sequenceMathworldPlanetmath of sets A1,A2, generating 𝒢. Note that if we let 𝒢n be the σ-algebra generated by A1,A2,,An, then 𝒢n is an increasing sequence of finite sub-σ-algebras such that n𝒢n generates 𝒢. The following result is general enough to apply in many useful cases, such as with the Boral σ-algebra on n.

Lemma 2.

Let G be a separable sub-σ-algebra of F. Then, the Radon-Nikodym derivative XG exists. If furthermore, Gn is an increasing sequence of finite σ-algebras satisfying G=σ(nGn) then EP[|XG-XGn|]0 as n.


Let us set XnX𝒢n. If m<n then the conditional expectation 𝔼[Xn𝒢m] is 𝒢m-measurable, and for every A𝒢m,


This equality just uses the definition of the conditional expectation and then the definition of Xn as the Radon-Nikodym derivative restricted to 𝒢n. So, 𝔼[Xn𝒢m] is the Radon-Nikodym derivative restricted to 𝒢m, and equals Xm (almost-surely).

Therefore, Xn is a martingale and the martingale convergence theorem implies that the limit

X𝒢=limnXn (1)

exists almost surely. We now show that the sequence Xn is uniformly integrable. Choose any ϵ>0. As is absolutely continuous with respect to , there exists a δ>0 such that (A)<ϵ whenever (A)<δ. Using


we see that (Xn>K)<δ whenever K>δ-1 and, therefore, (Xn>K)<ϵ. So


for every n, showing that Xn is a uniformly integrable sequence with respect to . Therefore, convergence in (1) is in L1, and 𝔼[|Xn-X𝒢|]0 as n. So, for any An𝒢n,

𝔼[X𝒢1A]=limm𝔼[Xm1A]=(A). (2)

By linearity and the monotone convergence theoremMathworldPlanetmath, the collectionMathworldPlanetmath of sets A satisfying (2) is a Dynkin system containing the π-system n𝒢n so, by Dynkin’s lemma, is satisfied for every Aσ(n𝒢n)=𝒢 and, by definition, X𝒢 is the Radon-Nikodym derivative restricted to 𝒢. ∎

Finally, by approximating by finite σ-algebras we can prove the Radon-Nikodym theorem for arbitrary inseparable σ-algebras .

Proof of the Radon-Nikodym theorem:

First, we use contradictionMathworldPlanetmathPlanetmath to show that for any ϵ>0 there exists a finite σ-algebra 𝒢 satisfying 𝔼[|X𝒢-X|]<ϵ for every finite σ-algebra with 𝒢F. If this were not the case, then by inductionMathworldPlanetmath we could find an increasing sequence of finite sub-σ-algebras of satisfying 𝔼[|X𝒢n-X𝒢m|]ϵ. However, letting 𝒢=σ(n𝒢n), Lemma 2 shows that X𝒢 exists and


— a contradiction.

So, there exists a sequence of finite sub-σ-algebras 𝒢n of such that 𝔼[|X𝒢n-X|]<2-n for every finite sub-σ-algebra of containing 𝒢n. Let 𝒢 be the (separable) σ-algebra generated by n𝒢n, and set 𝒢~n=σ(k=1n𝒢k). By Lemma 2, the Radon-Nikodym derivative restricted to 𝒢, X𝒢, exists, and we show that it is the required derivativePlanetmathPlanetmath of with respect to .

Choose any set A and let n be the (finite) σ-algebra generated by 𝒢n{A}. Then, Xn exists and satisfies 𝔼[Xn1A]=(A) and,


So, 𝔼[X𝒢1A]=(A) as required.


  • 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
Title martingale proof of the Radon-Nikodym theorem
Canonical name MartingaleProofOfTheRadonNikodymTheorem
Date of creation 2013-03-22 18:34:10
Last modified on 2013-03-22 18:34:10
Owner gel (22282)
Last modified by gel (22282)
Numerical id 9
Author gel (22282)
Entry type Proof
Classification msc 60G42
Classification msc 28A15
Related topic RadonNikodymTheorem
Related topic MartingaleConvergenceTheorem