Doob submartingale convergence theorem

Fix rationals $a<b$. By F1, $(b-a)\mathbb{E}{U}_N[a,b]\le \mathbb{E}(X_N-a{)}^{+}\le C+\mathrm{\mid }a\mathrm{\mid }\mathrm{.}$ As $U_N\uparrow U_{\mathrm{\infty }}$, F2 gives $\mathbb{E}{U}_{\mathrm{\infty }}[a,b]<\mathrm{\infty }$. Hence $U_{\mathrm{\infty }}[a,b]<\mathrm{\infty }$ almost surely.

Intersect these probability-one events over the countable set $\{(a,b)\in {\mathbb{Q}}^2:a<b\}$. F4 makes the intersection have probability one. On it, if $\mathrm{lim\hspace{0.17em}inf}{X}_n<\mathrm{lim\hspace{0.17em}sup}{X}_n$, density supplies rationals $a<b$ strictly between them; the path then completes infinitely many upcrossings of $[a,b]$, contradicting step 1.1. Thus $X_n$ has an extended-real limit.

The submartingale property gives $\mathbb{E}{X}_n\ge \mathbb{E}{X}_0$. Therefore $\mathbb{E}{X}_n^{-}=\mathbb{E}{X}_n^{+}-\mathbb{E}{X}_n\le C-\mathbb{E}{X}_0\mathrm{.}$ Fatou applied separately to $X_n^{+}$ and $X_n^{-}$ shows that neither $+\mathrm{\infty }$ nor $-\mathrm{\infty }$ can occur on a positive-measure set and that $\mathbb{E}\mathrm{\mid }{X}_{\mathrm{\infty }}\mathrm{\mid }\le C+(C-\mathbb{E}{X}_0)<\mathrm{\infty }\mathrm{.}$ Taking the limit on the full-measure event and defining it arbitrarily on its null complement yields the claimed finite integrable random variable. AC has only the inherited use in F5.