Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $(X_n)_n$, $X$ random variables on this space. Consider a sub $\sigma$-algebra $\mathcal{G} \subseteq \mathbb{F}$.
Suppose $X_n \to X$ almost surely and that $|X_n| \leq Y$ for all $n \geq 1$ where $Y$ is a positive, integrable random variable.
Is it true that $$\mathbb{E}[X_n \mid \mid \mathcal{G}] \to
\mathbb{E}[X\mid\mid \mathcal{G}]$$ in $L_1$?
My attempt:
I already know that $|\mathbb{E}[X_n \mid \mid \mathcal{G}] – \mathbb{E}[X\mid\mid \mathcal{G}]| \to 0$ a.s. (from another proof).
However, we also know that for $n \geq 1$, with probability $1$,
$$|\mathbb{E}[X_n \mid \mid \mathcal{G}] – \mathbb{E}[X\mid\mid \mathcal{G}]| \leq 2 \mathbb{E}[|Y| \mid \mid \mathcal{G}]$$ and $\mathbb{E}[|Y| \mid \mid \mathcal{G}]$ is integrable, so by the dominated convergence theorem:
$$\mathbb{E}[|\mathbb{E}[X_n \mid \mid \mathcal{G}] – \mathbb{E}[X\mid\mid \mathcal{G}]|] \to \mathbb{E}[0] = 0$$
and thus we have convergence in $L_1$.
Is this correct?
Best Answer
Using the conditional Jensen's inequality, $$ \mathsf{E}|\mathsf{E}[X_n\mid\mathcal{G}]-\mathsf{E}[X\mid\mathcal{G}]|\le \mathsf{E}|X_n-X|\to 0. $$