Considering a right-continuous sub-martingale $X$ bounded in $L^1,$ in theorem $33,$ it's possible to write $X$ as the difference of two positive right-continuous super-martingales.
The proof refers to $9$ (picture) for a proof, to obtain right-continuity he wrote: "taking right limits along the rationals".
How to explain this ? I tried to take $Y'_u =\inf_{r \in \mathbb{Q} \cap ]u,\infty[} Y_r$ (Helly theorem uses this form to obtain a right-continuous function), in this case $Y_u'$ is not super-martingale (at least it's not measurable $\mathcal{F}_u$).
You could take $Y'_t:=\liminf_{r\downarrow t,r\in\Bbb Q}Y_r$. This process is everywhere defined, adapted, and $Y'_t=\lim_{s\downarrow t}Y_s$ for all $t\ge 0$ a.s.