preservation of uniform integrability
Let $(\mathrm{\Omega},\mathcal{F},\mathbb{P})$ be a measure space^{}. Then, a uniformly integrable set $S$ of measurable functions^{} $f:\mathrm{\Omega}\to \mathbb{R}$ will remain uniformly integrable if it is enlarged by various operations^{}, such as taking convex combinations and conditional expectations. The following theorem lists some of the operations which preserve uniform integrability.
Theorem.
Suppose that $S$ is a bounded and uniformly integrable subset of ${L}^{\mathrm{1}}$. Let ${S}^{\mathrm{\prime}}$ be the smallest set containing $S$ such that all of the following conditions are satisfied. Then, ${S}^{\mathrm{\prime}}$ is also a bounded and uniformly integrable subset of ${L}^{\mathrm{1}}$.

1.
${S}^{\prime}$ is absolutely convex. That is, if $f,g\in {S}^{\prime}$ and $a,b\in \mathbb{R}$ are such that $a+b\le 1$ then $af+bg\in {S}^{\prime}$.

2.
If $f\in {S}^{\prime}$ and $g\le f$ then $g\in {S}^{\prime}$.

3.
${S}^{\prime}$ is closed under convergence in measure^{}. That is, if ${f}_{n}\in {S}^{\prime}$ converge in measure to $f$, then $f\in {S}^{\prime}$.

4.
If $f\in {S}^{\prime}$ and $\mathcal{G}$ is a sub$\sigma $algebra of $\mathcal{F}$ such that ${\mu }_{\mathcal{G}}$ is $\sigma $finite, then the conditional expectation ${\mathbb{E}}_{\mu}[f\mid \mathcal{G}]$ is in ${S}^{\prime}$.
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 $\mathrm{\Phi}:\mathbb{R}\to [0,\mathrm{\infty})$ such that $\mathrm{\Phi}(x)/x\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$ and
$$\int \mathrm{\Phi}(f)\mathit{d}\mu $$ 
is bounded over all $f\in S$ (see equivalent conditions for uniform integrability). Suppose that it is bounded by $K>0$. Also, by replacing $\mathrm{\Phi}$ by $\mathrm{\Phi}(x)+x$ if necessary, we may suppose that $\mathrm{\Phi}(x)\ge x$. Then, let $\overline{S}$ be
$$\overline{S}=\{f\in {L}^{1}:\int \mathrm{\Phi}(f)\mathit{d}\mu \le K\},$$ 
which is a bounded and uniformly integrable subset of ${L}^{1}$ containing $S$. To prove the result, it just needs to be shown that $\overline{S}$ is closed under each of the operations listed above, as that will imply ${S}^{\prime}\subseteq \overline{S}$.
First, the convexity and symmetry of $\mathrm{\Phi}$ gives
$$\int \mathrm{\Phi}(af+bg)\mathit{d}\mu \le \int \left(a\mathrm{\Phi}(f)+b\mathrm{\Phi}(g)\right)\mathit{d}\mu =a\int \mathrm{\Phi}(f)\mathit{d}\mu +b\int \mathrm{\Phi}(g)\mathit{d}\mu \le K$$ 
for any $f,g\in \overline{S}$ and $a,b\in \mathbb{R}$ with $a+b\le 1$. So, $af+bg\in \overline{S}$. Similarly, if $g\le f$ and $f\in \overline{S}$ then $\mathrm{\Phi}(g)\le \mathrm{\Phi}(f)$ and, $g\in \overline{S}$.
Now suppose that ${f}_{n}\in \overline{S}$ converge in measure to $f$. Then Fatou’s lemma gives,
$$\int \mathrm{\Phi}(f)\mathit{d}\mu \le \underset{n\to \mathrm{\infty}}{lim\; inf}\int \mathrm{\Phi}({f}_{n})\mathit{d}\mu \le K$$ 
so, $f\in \overline{S}$.
Finally suppose that $f\in \overline{S}$ and $g={\mathbb{E}}_{\mu}[f\mid \mathcal{G}]$. Using Jensen’s inequality,
$$\int \mathrm{\Phi}(g)d\mu \le \int {\mathbb{E}}_{\mu}[\mathrm{\Phi}(f)\mid \mathcal{G}]d\mu =\int \mathrm{\Phi}(f)d\mu \le K,$$ 
so $g\in \overline{S}$.
Title  preservation of uniform integrability 

Canonical name  PreservationOfUniformIntegrability 
Date of creation  20130322 18:40:20 
Last modified on  20130322 18:40:20 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  4 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 28A20 