I need to prove the following statement.
Let $X$ be a non negative martingale such that $X_n\rightarrow 0$ a.s. when $n\rightarrow \infty$. Define $X^*=supX_n$. Prove that for all $x>0$
$$P[X^* \geq x | \mathcal{F}_0]= 1 \wedge X_0 / x$$
I think I've got the easy case
- if $x\leq X_0$
Then necessarily $x\leq X^*$ for the sup property. Then it follows that for $1\leq X_0 /x$ we have that $P[X^* \geq x | \mathcal{F}_0]= 1$. But I can't figure out the other case.