 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.