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 $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{F}\mathrm{,}\mathrm{P}\mathrm{)}$. Then, the set

$$\{𝔼[X\mid 𝒢]:X\in S\mathit{\text{ and }}𝒢\mathit{\text{ is a sub-}}\sigma \mathit{\text{-algebra of }}ℱ\}$$

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 $𝔼[|X|]\le L$ for every $X\in S$.Also, choosing any $ϵ>0$, there is a $\delta >0$ so that

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

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

$$|Y|\le 𝔼[|X|\mid 𝒢].$$

So, applying Markov’s inequality,

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

and, therefore