Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable
Its is clear that not all supermartingales have a right continuous versions since the deterministic supermartingale $X=\mathbb{1}_{[0,1]}$ . So these must be something special about this supermartingale but I cant show it. Can someone help me out?