In my course on Stochastic Calculus,a closed submartingale was defined
Def:We say that a submartingale $(X_t)_{t \in \mathbb{R_+}}$ is closed by $X_{\infty}$ if there exists a random variable $X_{\infty}$ that is $\mathcal{F}_{\infty}= \sigma(\mathcal{F}_{t},t \geq 0)$ measurable and integrable such that for every t we have $$X_t \leq E[X_{\infty} \vert \mathcal{F_t}] \text{a.s.}$$
I understand the definition which is very clear but I do not get the motivation behind it . If someone could explain it would be really helpful.