Expectation of supremum Let $x(t)$ a real valued stochastic process and $T>0$ a constant. Is it true that:
$$\mathbb{E}\left[\sup_{t\in [0,T]} |x(t)|\right] \leq T \sup_{t\in [0,T]} \mathbb{E}\left[|x(t)|\right] \text{ ?}$$
Thanks for your help.

Elaboration on the comment by Zhen, just consider $x(t) = 1$ a.s. for all $t$ and $T = 0.5$
