I'm trying to understand, why a stopped submartingale is again a submartingale. In the lecture notes to my lecture this is just stated as a corollary of Doob's Optional Sampling Theorem but I don't see why the stopped submartingale is necessarily integrable. More explicitely, I'm trying to prove:
Let $X$ be a (continuous time) cadlag submartingale, $T$ a stopping time, $t\geq 0$.
Then, $\mathbb E \left|X^T_t\right|=\mathbb E\left|X_{\min\{T,t\}}\right|<\infty$.
Thanks for any advice!