Reverse filtration and reverse martingale

Assume AC. A reverse filtration is a decreasing sequence ${\mathcal{G}}_0\supseteq {\mathcal{G}}_1\supseteq \cdots$ of sub-$\sigma$-algebras, and ${\mathcal{G}}_{\mathrm{\infty }}:={\bigcap }_{n\ge 0}{\mathcal{G}}_n$. An integrable process $X_n$, with $X_n$ measurable for ${\mathcal{G}}_n$, is a reverse martingale if $\mathbb{E}[X_m\mid {\mathcal{G}}_n]=X_n\text{a.s. whenever }m\le n\mathrm{.}$

It is enough to require the adjacent identities $\mathbb{E}[X_n\mid {\mathcal{G}}_{n+1}]=X_{n+1}$: iterating them with Tower property of conditional expectation gives the displayed multistep identity, while the latter immediately includes adjacent indices. In particular $X_n=\mathbb{E}[X_0\mid {\mathcal{G}}_n]$. AC is used for existence of the conditional expectations and, if concrete versions are required, selection of this countable family; all identities are identities of almost-everywhere classes.