Conditional-expectation convergence along a filtration

If ${\mathcal{F}}_{\infty }=\sigma ({\bigcup }_n{\mathcal{F}}_n)$ and $Z$ is bounded, thenalmost surely and in $L^1$. Testing the almost-sure martingale limit on the algebra ${\bigcup }_n{\mathcal{F}}_n$ identifies it with the terminal conditional expectation.

If bounded random variables $X_n$ converge almost surely to $X$, thenalmost surely and in $L^1$. For $Z_n={\sup }_{m\ge n}|X_m-X|$, the variables $\mathbb{E}[Z_n\mid {\mathcal{F}}_n]$ form a nonnegative supermartingale with expectations tending to zero.