# conditional expectations are uniformly integrable

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

###### Theorem.

Let $S$ be a uniformly integrable set of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then, the set

 $\left\{\mathbb{E}[X\mid\mathcal{G}]:\textrm{X\in S and \mathcal{G} is a % sub-\sigma-algebra of \mathcal{F}}\right\}$

is also uniformly integrable.

To prove the result, we first use the fact that uniform integrability implies that $S$ is $L^{1}$-bounded. That is, there is a constant $L>0$ such that $\mathbb{E}[|X|]\leq L$ for every $X\in S$. Also, choosing any $\epsilon>0$, there is a $\delta>0$ so that

 $\mathbb{E}[|X|1_{A}]<\epsilon$

for all $X\in S$ and $A\in\mathcal{F}$ with $\mathbb{P}(A)\leq\delta$.

Set $K=L/\delta$. Then, if $Y=\mathbb{E}[X\mid\mathcal{G}]$ for any $X\in S$ and $\mathcal{G}\subseteq\mathcal{F}$, Jensen’s inequality gives

 $|Y|\leq\mathbb{E}[|X|\mid\mathcal{G}].$

So, applying Markov’s inequality,

 $\mathbb{P}(|Y|>K)\leq K^{-1}\mathbb{E}[|Y|]\leq K^{-1}\mathbb{E}[|X|]\leq L/K=\delta$

and, therefore

 $\mathbb{E}[|Y|1_{\{|Y|>K\}}]\leq\mathbb{E}[|X|1_{\{|Y|>K\}}]<\epsilon.$
Title conditional expectations are uniformly integrable ConditionalExpectationsAreUniformlyIntegrable 2013-03-22 18:40:08 2013-03-22 18:40:08 gel (22282) gel (22282) 5 gel (22282) Theorem msc 28A20 msc 60A10 ConditionalExpectation UniformlyIntegrable