A dyadic martingale converges to the original L1 variable

Every positive-length level-$n$ cell is the union of its two level-$(n+1)$ children, while the singleton $\{0\}$ is itself a generator at every level. Hence ${\mathcal{F}}_n\subseteq {\mathcal{F}}_{n+1}$. Dyadic half-open intervals form a countable base for the relative topology of $[0,1]$: every open interval is the countable union of dyadic cells whose closures lie inside it, with endpoints handled by the stated convention. Thus F1 gives $\sigma ({\bigcup }_n{\mathcal{F}}_n)=\mathcal{B}$.

F2 gives $M_n\to \mathbb{E}[X\mid \mathcal{B}]$ almost surely and in $L^1$. Since $X$ is $\mathcal{B}$-measurable, F3 identifies the latter conditional expectation with $X$. The singleton endpoint convention changes no $L^1$ or almost-sure statement. AC is used exactly through F4.