The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 \leq t \leq T}$ with $L_0 = 0$ on the probability space $(\Omega, \mathcal{F},P)$, is it possible to approximate it uniformly on $[0,T]$ on the dyadics by step functions, i.e. if $D_n=2^{-n} T \mathbb{N} \cap [0,T]$ is the set of all $n$-th dyadic points, do I have the following convergence $$ \sup_{t \in [0,T]}\left|L_t(\omega) - \sum_{t_i \in D_n}L_{t_i}(\omega) 1_{((t_i,t_i+1]]}(t, \omega)\right| \rightarrow 0, \text{ for } n \rightarrow \infty $$ for almost every $\omega \in \Omega$ ? The function $1_{((t_i,t_{i+1}]]}$ denotes the left half open stochastic interval.
Thanks a lot in advance!