Extension of result to stopping times

Question

Suppose I have a UI martingale $M_t$. Then by the martingale convergence theorem, I know that there is $M_\infty$ such that $\mathbb{E}[M_\infty\vert\mathcal{F}_t]=M_t$. I want to extend the result to stopping times $\tau$, i.e. $\mathbb{E}[M_\infty\vert\mathcal{F}_\tau]=M_\tau$ without using Doob's optional sampling theorem (in fact I want to use the result for a prove of the optional sampling theorem.) Any help is appreciated!

Answer 1

So start by proving the finite time statement:

$$ \mathbb E[M_n \mid \mathcal F_{\tau} ] = M_{\tau \wedge n}$$ We then try and prove the statement by taking $n \to \infty$. Argue that $\mathbb E[M_n \mid \mathcal F_{\tau}] \to \mathbb E[M_{\infty} \mid \mathcal F_{\tau}]$ in $L^1$ (and hence a subsequence does so almost surely) and that $M_{\tau \wedge n} \to M_{\tau}$ almost surely to complete the argument. If you have any trouble with these steps please say.