= Solution
A <càdlàg martingale> is an <adapted process> $X$ with <càdlàg> paths, integrable $X_t$ at each finite time, and
$$
\boxed{\mathbb E[X_t\mid\mathcal F_s]=X_s\quad(0\leq s\leq t).}
$$
The <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 \b[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
$$
\boxed{\mathcal F_t=\bigcap_{u>t}\mathcal F_u.}
$$
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