# preservation of uniform integrability

###### Theorem.

Suppose that $S$ is a bounded and uniformly integrable subset of $L^{1}$. Let $S^{\prime}$ be the smallest set containing $S$ such that all of the following conditions are satisfied. Then, $S^{\prime}$ is also a bounded and uniformly integrable subset of $L^{1}$.

1. 1.

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

2. 2.

If $f\in S^{\prime}$ and $|g|\leq|f|$ then $g\in S^{\prime}$.

3. 3.

$S^{\prime}$$f_{n}\in S^{\prime}$ converge in measure to $f$, then $f\in S^{\prime}$.

4. 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 $\Phi\colon\mathbb{R}\rightarrow[0,\infty)$ such that $\Phi(x)/|x|\rightarrow\infty$ as $|x|\rightarrow\infty$ and

 $\int\Phi(f)\,d\mu$

is bounded over all $f\in S$ (see equivalent conditions for uniform integrability). Suppose that it is bounded by $K>0$. Also, by replacing $\Phi$ by $\Phi(x)+|x|$ if necessary, we may suppose that $\Phi(x)\geq|x|$. Then, let $\bar{S}$ be

 $\bar{S}=\left\{f\in L^{1}:\int\Phi(f)\,d\mu\leq K\right\},$

which is a bounded and uniformly integrable subset of $L^{1}$ containing $S$. To prove the result, it just needs to be shown that $\bar{S}$ is closed under each of the operations listed above, as that will imply $S^{\prime}\subseteq\bar{S}$.

First, the convexity and symmetry of $\Phi$ gives

 $\int\Phi(af+bg)\,d\mu\leq\int\left(|a|\Phi(f)+|b|\Phi(g)\right)\,d\mu=|a|\int% \Phi(f)\,d\mu+|b|\int\Phi(g)\,d\mu\leq K$

for any $f,g\in\bar{S}$ and $a,b\in\mathbb{R}$ with $|a|+|b|\leq 1$. So, $af+bg\in\bar{S}$. Similarly, if $|g|\leq|f|$ and $f\in\bar{S}$ then $\Phi(g)\leq\Phi(f)$ and, $g\in\bar{S}$.

Now suppose that $f_{n}\in\bar{S}$ converge in measure to $f$. Then Fatou’s lemma gives,

 $\int\Phi(f)\,d\mu\leq\liminf_{n\rightarrow\infty}\int\Phi(f_{n})\,d\mu\leq K$

so, $f\in\bar{S}$.

Finally suppose that $f\in\bar{S}$ and $g=\mathbb{E}_{\mu}[f\mid\mathcal{G}]$. Using Jensen’s inequality,

 $\int\Phi(g)\,d\mu\leq\int\mathbb{E}_{\mu}[\Phi(f)\mid\mathcal{G}]\,d\mu=\int% \Phi(f)\,d\mu\leq K,$

so $g\in\bar{S}$.

Title preservation of uniform integrability PreservationOfUniformIntegrability 2013-03-22 18:40:20 2013-03-22 18:40:20 gel (22282) gel (22282) 4 gel (22282) Theorem msc 28A20