Non-negative martingale $X_n \rightarrow 0$ a.s. prove that $P[X^* \geq x | \mathcal{F}_0]= 1 \wedge X_0 / x$
Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$