uniformly integrable

Let μ be a positive measureMathworldPlanetmath on a measurable spaceMathworldPlanetmathPlanetmath. A collectionMathworldPlanetmath of functions {fα}𝐋1(μ) is uniformly integrable, if for every ϵ>0, there exists δ>0 such that

|Efα𝑑μ|<ϵwhenever μ(E)<δ, for any α.

(The absolute valuePlanetmathPlanetmathPlanetmathPlanetmath sign outside of the integral above may appear under the integral sign instead without affecting the definition.)

The usefulness of this definition comes from the Vitali convergence theorem, which uses it to characterize the convergence of functions in 𝐋1(μ).

Definition in probability theory

In probability , a different, and slightly stronger, definition of “uniform integrability”, is more commonly used:

A collection of functions {fα}𝐋1(μ) is uniformly integrable, if for every ϵ>0, there exists t0 such that

[|fα|t]|fα|𝑑μ<ϵfor every α.

Assuming μ is a probability measure, this definition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the previous one together with the condition that |fα|𝑑μ is uniformly bounded for all α.


  1. 1.

    If a finite number of collections are uniformly integrable, then so is their finite union.

  2. 2.

    A single f𝐋1(μ) is always uniformly integrable.

    To see this, observe that f must be almost everywhere non-infinite. Thus f1[|f|>k] goes to zero a.e. as k, and it is boundedPlanetmathPlanetmathPlanetmath by |f|. Then [|f|>k]|f|𝑑μ0 by the dominated convergence theorem. Choosing k big enough so that [|f|>k]|f|𝑑μ<ϵ, and letting δ=ϵ/k, we have, when μ(E)<δ,



  1. 1.

    If g is an integrable function, then the collection consisting of all measurable functionsMathworldPlanetmath f dominated by g — that is, |f|g — is uniformly integrable.

  2. 2.

    If X is a 𝐋1 random variableMathworldPlanetmath on a probability space Ω, then the set of all of its conditional expectations,

    {𝔼[X𝒢]:𝒢 is a σ-algebra of Ω},

    is always uniformly integrable.

  3. 3.

    If there is an unboundedPlanetmathPlanetmath increasing function ϕ:[0,)[0,) such that


    is uniformly bounded for all α, then the collection {fα} is uniformly integrable.


  • 1 Kai Lai Chung. A Course in Probability Theory, third ed. Academic Press, 2001.
  • 2 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
  • 3 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
Title uniformly integrable
Canonical name UniformlyIntegrable
Date of creation 2013-03-22 15:22:55
Last modified on 2013-03-22 15:22:55
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 23
Author stevecheng (10074)
Entry type Definition
Classification msc 28A20
Synonym uniform integrability
Synonym uniform absolute continuity
Related topic VitaliConvergenceTheorem
Related topic ConditionalExpectationsAreUniformlyIntegrable