Representation Theorem for functionals of Continuous Semimartingales

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable. Does it mean that $\xi = F(X_u : 0 \leq u \leq T)$ for some measurable $F : C[0,T] \rightarrow \mathbb{R}$ ? Here the topology on $C[0,T]$ is generated by cylindrical sets. Intuitively the answer seems to be yes. I attempted to prove it by assuming that $\xi$ only depends on $X$ at finite number of times. However, I have problems passing to the limit. Any help/references is highly appreciated.

Answer 1

Fix $T>0$ and let $\zeta = (X_t)_{t\le T}$, it then holds that $\zeta$ is a measurable mapping from $\Omega$ to $C[0,T]$. By conventional measure theory results, we then also obtain that $\mathcal{F}_T^X$ is generated by the single variable $\zeta$, $\mathcal{F}^X_T = \sigma(\zeta)$. Therefore, by the Doob-Dynkin lemma (see the first lemma of Section A.IV.3 of "Classical potential theory and its probabilistic counterpart" by J. L. Doob), it holds that there exists a measurable mapping $F:C[0,T]\to\mathbb{R}$ with the property that $\xi =F(\zeta) = F((X_t)_{t\le T})$, as required.