Define the predictable $\sigma$-algebra as $$ \mathcal P := \sigma(X: \text{ $X$ is a left-continuous and adapted process }), $$ and say that a stochastic process is predictable if it is measurable w.r.t. $\mathcal P$. [The stochastic processes are assumed to take values in some Polish space $E$ and be indexed by $[0,\infty)$]
As measurablility is preserved under pointwise limits, a pointwise limit of left-continuous and adapted processes will be predictable.
Is it true that any predictable process is a pointwise limit (or even a.s. pointwise limit if we assume our probability space is complete) of a sequence of left-continuous and adapted processes.?
More generally, is it always true that if we define $\mathcal F := \sigma(f: \text{ $ f $ with some property })$, then every $\mathcal F $-measurable function is the limit of a sequence of functions with that same property?