equivalent conditions for uniform integrability

Let (Ω,,μ) be a measure spaceMathworldPlanetmath and S be a boundedPlanetmathPlanetmathPlanetmath subset of L1(Ω,,μ). That is, |f|𝑑μ is bounded over fS. Then, the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

  1. 1.

    For every ϵ>0 there is a δ>0 so that


    for all fS and μ(A)<δ.

  2. 2.

    For every ϵ>0 there is a K>0 satisfying


    for all fS.

  3. 3.

    There is a measurable functionMathworldPlanetmath Φ:[0,) such that Φ(x)/|x| as |x| and


    is bounded over all fS. Moreover, the function Φ can always be chosen to be symmetricPlanetmathPlanetmath and convex.

So, for bounded subsets of L1, either of the above properties can be used to define uniform integrability. Conversely, when the measure space is finite, then conditions (2) and (3) are easily shown to imply that S is bounded in L1.

To show the equivalence of these statements, let us suppose that |f|𝑑μ<L for fS.

(1) implies (2)

For ϵ>0, property (1) gives a δ>0 so that A|f|𝑑μ<ϵ whenever fS and μ(A)<δ. Choosing K>L/δ, Markov’s inequalityMathworldPlanetmath gives


and, therefore, |f|>K|f|𝑑μ<ϵ.

(2) implies (3)

For each n=1,2,, property (2) gives a Kn satisfying


Without loss of generality, the Kn can be chosen to be increasing to infinityMathworldPlanetmathPlanetmath, so we can define Φ(x)=n(|x|-Kn)+. Then,


(3) implies (1)

First, suppose that Φ(f)𝑑μ<M for fS. For ϵ>0, the condition that Φ(x)/|x| as |x| gives a K>0 such that Φ(x)/|x|2M/ϵ whenever |x|>K. Setting δ=ϵ/2K,


whenever μ(A)<δ and fS.

Title equivalent conditions for uniform integrability
Canonical name EquivalentConditionsForUniformIntegrability
Date of creation 2013-03-22 18:40:17
Last modified on 2013-03-22 18:40:17
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 28A20