Doob's maximal inequality and probability of the sup

Question

Referring to the Doob's Maximal inequality we have that letting $M_t$ be a continous martingale, $\lambda >0$, $p >1$ and $T >0$ we have $P(\sup_{0 \leq t \leq T} |M_t| > \lambda) \leq \frac{1}{\lambda^p} \mathbb{E}[|M_T|^p]$. I did not understand if the sup is the sup of the events ($|M_t| > \lambda$ : $t \in [0,T]$) or if it is the sup of the random variable $|M_t|$ itself.

Answer 1

Let $M_T^* = \sup_{0\le t\le T}|M_t|$. Then the claim is that $P(M_T^*>\lambda)\le\lambda^{-p}\mathbb E[|M_T|^p]$. In other words, it is the sup of the random variables $|M_t|$ for $0\le t\le T$.