In discrete-time, we say that a process X defined in $\left(\Omega, \mathscr{F}, \lbrace\mathscr{F}_n\rbrace, \mathbb{P}\right)$ given by X $ = \lbrace X_n, n \in \mathbb{N}\rbrace$ is predictable if $ \forall n \in \mathbb{N}, X_n$ is $\mathscr{F}_{n-1}$-measurable. An intuitive meaning of this is that, one can predict the value of $X$ at time $n$ despite only having the information at time $n-1$.
However, as defined by Revuz and Yor, for continuous time, a process $X$ is predictable if the map $\left(\omega, t\right) \to X_t(\omega)$ is measurable with respect to the $\sigma$-field generated by the space of adapted processes which are left continuous on $\left(0,\infty\right)$.
Can someone enlighten me on the similarity of the two definitions? And how would the "predictability" of $X$ be intuitive from the definition in continuous time (similar to the intuitive meaning in discrete-time)?