If $Y_n=E(X|\mathcal{F}_n)$ for some $\mathcal{F}_n \subset \mathcal{F}$ and if $Y_n \to Y$ with probability one. Show $Y_n \to Y$ in $L^1$.
I am trying to show that $Y_n$ is uniformly integrable (UI). If it is known that $E|X|<\infty$, then there is a theorem that tells us that there exists a convex function $\phi$ such that $\phi(x)/x \to_x \infty$ and $E[\phi(|X|)]<\infty$. This should help us show that $Y_n$ are UI with Jensen's inequality. But how should I show $E|X|<\infty$ or is there other ways?