I am trying to self study chapter 4.7 Backwards Martingales of Durret's <Probability: Theory and Examples> and encountered a small obstacle just now.
A backwards martingale is a martingale indexed by the negative integers, i.e., $X_n$, $n\leq 0$, adapted to an increasing sequence of $\sigma$-fields $\mathcal{F}_n$ with $E(X_{n+1}|\mathcal{F}_n)=X_n \ for \ n\leq -1$. I know that $X_{-\infty}=\lim_{n\to -\infty}X_n$ exists a.s. and in $L^1$. Then, I have a puzzle about the following theorem.
If $X_{-\infty}=\lim_{n\to -\infty}X_n$ and $\mathcal{F}_{-\infty}=\cap_n\mathcal{F}_n$, then $ X_{-\infty}=E(X_0|\mathcal{F}_n)$.
The proof of the theorem states "Clearly, $X_{-\infty}\in\mathcal{F}_n$", but why. The following is my try.
- Of course we cannot use the usual theorem that the limit of measurable functions is measurable since $X_1\notin \mathcal{F_{-\infty}}$.
- I tried to prove in the following way $$\{X_{-\infty}<x\} =\{\lim_{n\to -\infty}x\}\\ =\{\limsup_{n\to -\infty}<x\}\\ =\bigcup_{1\leq k,n<\infty}\bigcap_{j=k}^{\infty}\{X_j <c-1/n\}$$, where the second equality is because $X_n$ converges a.s. and the third equality is because $X_{-\infty}<x$ iif there are natural numbers n and k for which $X_j < x-1/n$ for all $j\geq k$.
Then how should I proceed? Of course $\bigcap_{j=k}^{\infty}\{X_j <c-1/n\} \in \mathcal{F}_1,\mathcal{F}_2,\mathcal{F}_3,..\mathcal{F}_k$, but I cannot prove it belongs to $\mathcal{F}_{-\infty}$. I will be extremely grateful if you can provide any suggestions.
$X_k, X_{k+1},...$ are all measurable w.r.t $\mathcal F_n$ and $\{X_k, X_{k+1},... \}\to X$ a.s . Hence $X$ is measuarable w.r.t. $\mathcal F_n$.