I do have problems to prove the following, without even being sure whether the statement is true:
Let $t\in[0,T]$ and $(X_s)_{s\in [t,T]}$ be a continuous stochastic process on some filtered prob space satisfying the usual conditions such that $X_s\geq 0$ for all $s\in [t,T]$. Assume that $\mathbb{E}\left[\sup_{t\leq s\leq T} X_s\right]<\infty$. Then $\mathbb{E}[X_{\tau}]<\infty$ for every $[t,T]$-valued stopping time $\tau$.
How to do that? I wasn't able to find anything without martingale properties.
Thanx for your help!
(There's a little confusion with your time indexing. I'll assume $\Bbb E[\sup_{0\le s\le T}X_s]<\infty$.)
Let $M(\omega) :=\sup_{0\le s\le T}X(\omega)$. Convince yourself that because $(X_\tau)(\omega) = X_{\tau(\omega)}(\omega)$, you must have $(X_\tau)(\omega)\le M(\omega)$ for all $\omega$.