 Solution

A càdlàg martingale is an adapted process $X$ with càdlàg paths, integrable $X_t$ at each finite time, andThe càdlàg requirement means that the chosen path version is right-continuous and has a finite left limit at every positive time. With a complete filtration, it suffices that this holds on one common event of probability one, since the process can be redefined on the exceptional null set.

The usual hypotheses for a filtration are completeness and right continuity. Completeness means that ${\mathcal{F}}_0$ contains every subset of every null event of the ambient probability space; right continuity means