I was reading a paper by Marc Yor for my thesis and in the statement of one of the theorem he mentions Consider the super martingale $\{P(L>u|\mathcal{F}_u),u \geq 0\}$ where $L$ is a random time i.e it is a measurable random variable taking values in $\mathbb{R}_+$. They consider a right-continuous version of this super-martingale
It is clear that right continuous versions do not exist in general for any super-martingale. I considered for example the deterministic supermartingale $Y_t=\mathbb{1}_{[0,1]}(t)$ for all $\omega$ and it is clearly impossible to construct a right continuous modification. How could I show that the super-martingale of the above form permits such a (right) continuous modification?
My idea is that since the super-martingale can be written as a function of conditional probability distribution(pathwise), it is right continuous . Am I correct ? Or is my argument flawed here? How can I show this otherwise?