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.

Another way of looking at Zhen's comment:
If $x(t)$ is measured in inches and $T$ in seconds, then the left side would be in inches, whereas the right side would be in seconds times inches.  So if we change the units of time from seconds to eons, the numerical value on the right gets smaller, but that on the left does not.  The numerical value on the right can be made as close to $0$ as desired by making the units of time big enough, whereas that on the left remains fixed at a positive number.
