Is every predictable process a pointwise limit of left-continuous, adapted processes?

Question

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?

Answer 1

I don't think the generalization is true because the property could be too narrow. For example, consider functions from $[0,1]$ to $[0,1]$ equipped with the Borel sigma algebra. If we let $\mathcal{F} := \sigma(f : f(x) = x \text{ for all $x \in [0,1]$})$ then $\mathcal{F}$ is the Borel sigma algebra on $[0,1]$, but the only function that can be obtained by limits of functions with that property is the identity.

Answer 2

Any pointwise limit of left-continuous adapted processes would have sample paths $t\mapsto X_t(\omega)$ that were, at worst, in the fourth Baire class. Thus a deterministic process $X_t(\omega)=f(t)$, for a Borel function $f$ of the fifth Baire class, is not the pointwise limit of the type you request.

The good news is that any argument you might be trying to make based on such a pointwise approximation "fact" can likely be made through the use of the functional monotone class theorem.