= Solution
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> gives
$$
X_n\longrightarrow X_\infty\quad\lambda\text{-almost surely},\qquad
\mathbb E X_\infty\leq\mu([0,1)).
$$
For fixed $n$, apply the <Conditional Fatou lemma> to the nonnegative variables $X_m$, $m\geq n$. The <martingale> property makes each $\mathbb E[X_m\mid\mathcal F_n]$ equal to $X_n$, hence
$$
\boxed{\mathbb E[X_\infty\mid\mathcal F_n]\leq X_n\quad\lambda\text{-almost surely}.}
$$
The 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.
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