Q:

Almost sure convergence of non-rising subalgebras sequence

I'm studying almost sure convergence and trying to prove that the following statement is true or false:

For every non-rising sequence $\sigma$-subalgebras $G\small{n}\subset F$ for every $A \in F$ $P(A|G\small{n}) \xrightarrow{a.s.} P(A|\cap {\tiny n \in N} \space G\small{n})$

I feel that it is true. But how to prove it?

A:

This is an immediate consequence of the backward martingale convergence theorem. Any probability textbook will prove this. Also see here.

