Suppose that $X=(X_t)_{t\geq 0}$ is a stochastic process and $(\mathcal{F}_t)_{t\geq 0}$ is the sigma field generated by $X$. Is there a result that says the following or something similar: $Y$ is $\mathcal{F}_t$-measurable if and only if $Y=f(X_s:s\leq t)$ for some measurable real function $f$.
The answer given here seems to be this statement. However, I would like a proof or citation.
Also, how should $f(X_s:s\leq t)$ be interpreted rigorously?