conditional expectations are uniformly integrable

The collectionMathworldPlanetmath of all conditional expectations of an integrable random variableMathworldPlanetmath forms a uniformly integrable set. More generally, we have the following result.


Let S be a uniformly integrable set of random variables defined on a probability spaceMathworldPlanetmath (Ω,F,P). Then, the set

{𝔼[X𝒢]:XS and 𝒢 is a sub-σ-algebra of }

is also uniformly integrable.

To prove the result, we first use the fact that uniform integrability implies that S is L1-bounded. That is, there is a constant L>0 such that 𝔼[|X|]L for every XS. Also, choosing any ϵ>0, there is a δ>0 so that


for all XS and A with (A)δ.

Set K=L/δ. Then, if Y=𝔼[X𝒢] for any XS and 𝒢, Jensen’s inequalityMathworldPlanetmath gives


So, applying Markov’s inequality,


and, therefore

Title conditional expectations are uniformly integrable
Canonical name ConditionalExpectationsAreUniformlyIntegrable
Date of creation 2013-03-22 18:40:08
Last modified on 2013-03-22 18:40:08
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 28A20
Classification msc 60A10
Related topic ConditionalExpectation
Related topic UniformlyIntegrable