I'm trying to understand when a backward martingale is convergent. I've obtained that if the backward martingale is convergent a.e., then it is convergent in $L^1$ to $E[M_1| F_{ \infty } ]$.
I'm having some trouble finding a result about convergence a.e., in other words i've not found some hypothesis that lead to convergence a.e.
I add the definition of backward martingale in order to explain my notation:
Let $(M_n) _{n \geq 1}$ a integrable process and $(F_n)_{n \geq 1}$ a family of $\sigma $-field non-increasing. $M_n$ is a backward martingale if $M_n$ is $F_n$-measurable and $E[M_n|F_{n+1}]=M_{n+1}$.
$F_{\infty}$ denotes $\bigcap _{n \geq 1} F_n$.