The following can be regarded as the generalization of the maximal inequality for time-continuous martingales:
Let$ (X_t)_{t \ge 0}$ be a supermartingale with right-continuous sample paths. Then, for any $t > 0$ and every $\lambda > 0$:
$$\lambda \mathbb{P}(\sup_{0 \le s \le t} |X_s| > \lambda) \le \mathbb{E}[|X_0|] + 2\mathbb{E}[|X_t|]$$
The proof relies on the fact that I can use the discrete-time version on $D \cap [0,t]$ where $D$ is some dense, countable subset of $\mathbb{R}_{+}$
with $0, t \in D$. Due to the right-continuity of the sample paths, we then have $\sup_{s \in D \cap [0,t]} |X_s| = \sup_{s \in [0,t]} |X_s|$. This is clear to me, yet I can't come up with an example of a supermartingale with non-right-continuous sample paths that does not fulfill this maximal inequality.