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单词 UniformlyIntegrable
释义

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, thereexists 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 functionsin 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, thereexists t0 such that

[|fα|t]|fα|𝑑μ<ϵfor every α.

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

Properties

  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 boundedPlanetmathPlanetmathPlanetmathby |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,

    E|f|𝑑μ=E[|f|k]|f|𝑑μ+E[|f|>k]|f|𝑑μkμ(E)+ϵ=2ϵ.

Examples

  1. 1.

    If g is an integrable function, thenthe collection consisting ofall 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

    |fα|ϕ(|fα|)𝑑μ

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

References

  • 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.
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更新时间:2025/5/4 16:04:19