Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual conditions, $W$ an $\mathbb{F}$-Brownian motion and $X$ a continuous $\overline{\mathbb{F}^W}$ - adapted process with $\overline{\mathbb{F}^W}$ being the completed filtration generated by $W$. We equip $C([0,\infty),\mathbb{R})$ with the sigma algebra generated by the projections.
In the script my professor claims that $X:\Omega \rightarrow C([0,\infty),\mathbb{R})$ is $\sigma(W)$ - measureable, but I don't see why this should be true since $X$ is only $\overline{\mathbb{F}^W}$ - and not $\mathbb{F}^W$ - adapted.
Remark: This is used for showing that a strong solution to an SDE is a function of $W$.
I would be grateful for any help.