$X$ is a backwards martingale with $X_0\in L^1 $
According to the convergence theorem:$X_{-n}\to X_{-\infty} $ a.s.
But how to get the conclusion that $X_{-\infty}$ is $\mathcal F_{-\infty} $ measurable?
Unlike the forwards martingale:$X_n\in\mathcal F_n\subset\mathcal F_{\infty}$ so the limit function is $\in\mathcal F_\infty$.
Here $X_{-n}\in\mathcal F_{-n}\supset\mathcal F_{-\infty}$ perhaps $X_{-n}\notin\mathcal F_{-\infty}$
As the pointwise limit of the sequence $(X_k)_{k\leqslant-n}$ which is measurable with respect to $\mathcal F_{-n}$, the random variable $X_{-\infty}$ is measurable with respect to $\mathcal F_{-n}$. This holds for every $n$, hence $X_{-\infty}$ is measurable with respect to $\bigcap\limits_n\mathcal F_{-n}=$ $______$.