A lot of the theorems in stochastic analysis are stated for cadlag processes (i.e. right continuous processes with left limits), but I am having a hard time seeing why the "left limits" part is important. It seems like for the most part just right continuity is enough, so I was wondering if anybody had a general explanation for why the assumption of left limits is usually included.
For a specific example, Proposition 2.3.5 in Revuz and Yor's "Continuous Martingales and Brownian Motion" states
A cadlag adapted process $X$ is a martingale if and only if for every bounded stopping time $T$ the random variable $X_T \in L^1$ and $\mathbb{E}[X_T] = \mathbb{E}[X_0]$.
The "only if" part comes from the optional stopping theorem, which did not include the assumption that $X$ is cadlag (because martingales have cadlag modifications anyway when the filtration satisfies the usual conditions). The proof for the converse direction is to fix $s < t$ and $A \in \mathcal F_s$ and define $T = t 1_{A^c} + s 1_A$ and use that $\mathbb{E}[X_t] = \mathbb{E}[X_T]$ to show $\mathbb{E}[X_t 1_A] = \mathbb{E}[X_s 1_A]$ and hence $\mathbb{E}[X_t | \mathcal F_s] = X_s$, but this also doesn't seem to use the left limits assumption. I originally thought it was to ensure $X$ is progressively measurable so that $X_T$ is measurable, but being right continuous and adapted is enough to conclude $X$ is progressively measurable so I'm still confused on why we need left limits.
