preservation of uniform integrability

Let (Ω,,) be a measure spaceMathworldPlanetmath. Then, a uniformly integrable set S of measurable functionsMathworldPlanetmath f:Ω will remain uniformly integrable if it is enlarged by various operationsMathworldPlanetmath, such as taking convex combinations and conditional expectations. The following theorem lists some of the operations which preserve uniform integrability.


Suppose that S is a bounded and uniformly integrable subset of L1. Let S be the smallest set containing S such that all of the following conditions are satisfied. Then, S is also a bounded and uniformly integrable subset of L1.

  1. 1.

    S is absolutely convex. That is, if f,gS and a,b are such that |a|+|b|1 then af+bgS.

  2. 2.

    If fS and |g||f| then gS.

  3. 3.

    S is closed under convergence in measurePlanetmathPlanetmath. That is, if fnS converge in measure to f, then fS.

  4. 4.

    If fS and 𝒢 is a sub-σ-algebra of such that μ|𝒢 is σ-finite, then the conditional expectation 𝔼μ[f𝒢] is in S.

To prove this we use the condition that the set S is uniformly integrable if and only if there is a convex and symmetric function Φ:[0,) such that Φ(x)/|x| as |x| and


is bounded over all fS (see equivalent conditions for uniform integrability). Suppose that it is bounded by K>0. Also, by replacing Φ by Φ(x)+|x| if necessary, we may suppose that Φ(x)|x|. Then, let S¯ be


which is a bounded and uniformly integrable subset of L1 containing S. To prove the result, it just needs to be shown that S¯ is closed under each of the operations listed above, as that will imply SS¯.

First, the convexity and symmetry of Φ gives


for any f,gS¯ and a,b with |a|+|b|1. So, af+bgS¯. Similarly, if |g||f| and fS¯ then Φ(g)Φ(f) and, gS¯.

Now suppose that fnS¯ converge in measure to f. Then Fatou’s lemma gives,

Φ(f)𝑑μlim infnΦ(fn)𝑑μK

so, fS¯.

Finally suppose that fS¯ and g=𝔼μ[f𝒢]. Using Jensen’s inequality,


so gS¯.

Title preservation of uniform integrability
Canonical name PreservationOfUniformIntegrability
Date of creation 2013-03-22 18:40:20
Last modified on 2013-03-22 18:40:20
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Theorem
Classification msc 28A20