proof of Vitali convergence theorem


Let f1,f2, be Lp-integrable functions on a measure spaceMathworldPlanetmath (X,μ), for 1p<. The following conditions are necessary and sufficient for fn to be a Cauchy sequenceMathworldPlanetmathPlanetmath in the Lp(X,μ) norm:

  1. (i)

    the sequenceMathworldPlanetmath fn is Cauchy in measure;

  2. (ii)

    the functions {|fn|p} are uniformly integrable; and

  3. (iii)

    for each ϵ>0, there is a set A of finite measure, with fn𝟏(XA)<ϵ for all n.


We abbreviate |fn-fm| by fmn.

Necessity of (i).

Fix t>0, and let Emn={fmnt}. Then

μ(Emn)1/p=1tt 1(Emn)1tfmn0,as m,n.
Necessity of (ii).

Select N such that fn-fN<ϵ when nN. The family {|f1|p,,|fN-1|p,|fN|p} is uniformly integrable because it consists of only finitely many integrable functions.

So for every ϵ>0, there is δ>0 such that μ(E)<δ implies fn𝟏(E)<ϵ for nN. On the other hand, for n>N,


for the same sets E, and thus the entire infiniteMathworldPlanetmathPlanetmath sequence {|fn|p} is uniformly integrable too.

Necessity of (iii).

Select N such that fn-fN<ϵ for all nN. Let φ be a simple functionMathworldPlanetmathPlanetmath approximating fN in 𝐋p norm up to ϵ. Then fn-φ<2ϵ for all nN. Let AN={φ0} be the supportMathworldPlanetmath of φ, which must have finite measure. It follows that

fn𝟏(XAN)=fn-fn𝟏(AN) fn-φ+φ-fn𝟏(AN)

For each n<N, we can similarly construct sets An of finite measure, such that fn𝟏(XAn)<4ϵ. If we set A=A1AN-1AN, a finite union, then A has finite measure, and clearly fn𝟏(XA)<4ϵ for any n.


We show fmn to be small for large m,n by a multi-step estimate:

fmn fmn𝟏(AEmn)+fmn𝟏(Emn)+fmn𝟏(XA).

Use condition (iii) to choose A of finite measure such that fn𝟏(XA)<ϵ for every n. Then fmn𝟏(XA)<2ϵ.

Let t=ϵ/μ(A)1/p>0, and Emn={fmnt}. By condition (ii) choose δ>0 so that fn𝟏(E)<ϵ whenever μ(E)<δ. By condition (i), take N such that if m,nN, then μ(Emn)<δ; it follows immediately that fmn𝟏(Emn)<2ϵ.

Finally, fmn𝟏(AEmn)tμ(A)1/p=ϵ, since fmn<t on the complement of Emn. Hence fmn<5ϵ for m,nN. ∎

Remark. In the statement of the theorem, instead of dealing with Cauchy sequences, we can directly speak of convergence of fn to f in 𝐋p and in measure. This variation of the theorem is easily proved, for:

  • a sequence convergesPlanetmathPlanetmath in 𝐋p if and only if it is Cauchy in 𝐋p;

  • a sequence that converges in measure is automatically Cauchy in measure;

  • a simple adaptation of the argumentMathworldPlanetmath shows that fnf in 𝐋p implies fnf in measure; and

  • the limit in measure is unique.

Title proof of Vitali convergence theorem
Canonical name ProofOfVitaliConvergenceTheorem
Date of creation 2013-03-22 17:31:04
Last modified on 2013-03-22 17:31:04
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 5
Author stevecheng (10074)
Entry type Proof
Classification msc 28A20