Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 1 / b

The martingale convergence theorem in the form needed here states that a nonnegative supermartingale with uniformly bounded expectations converges almost surely to a finite integrable limit. Here $\mathbb{E}{X}_n=\mu ([0,1))$ for all $n$, so it applies to $X_n$. The ordinary Fatou lemma givesFor fixed $n$, apply the Conditional Fatou lemma to the nonnegative variables $X_m$, $m\ge n$. The martingale property makes each $\mathbb{E}[X_m\mid {\mathcal{F}}_n]$ equal to $X_n$, henceThe inequality is simultaneous for all $n$ after removing a countable union of null sets. Equality of expectations in the limit has not been assumed; it can fail for a measure with a singular component.