Does a predictable process have countably many discontinuities Suppose $\mathbb{F}$ is a filtration on the complete probability space $(\Omega, \mathcal{F},\mathbb{P})$ and the function $V$ is $\mathbb{F}$-predictable. That means that $V$ is measurable w.r.t the sigma algebra generated by the left continuous functions.
I wonder whether this gives the opportunity to state something on the regularity of predictable functions, in particular:

*

*Is there a representation/modification of $V$ which consists of left-continuous functions?


*Can I state, that $V$ has only countable many jumps (either based on the first point or the fact that the sigma algebra is generated by left-continuous functions)?

The process $V_t=1_{\{0\}}(B_t)$ is optional (hence predictable) with respect to the filtration of the Brownian motion $B$, but $t\mapsto V_t$ is a.s. discontinuous at every point of the (a.s. perfect) set $\mathcal Z:=\{t\ge 0: B_t=0\}$.
