I don't know if this is a difficult question, but I have absolutely no intuition about it.
Let us consider a probability space $(\Omega, \mathcal F, \mathbb P)$ supporting a $\mathbb R-$valued stochastic process $(X_t)_{t\in[0,T]}$. Let us denote by $(\mathcal F^X_t)_{t\in [0,T]}$ the canonical filtration of $X$, i.e. $$\mathcal F^X_t = \sigma \left( X_s, s\le t \right).$$ Now, I know that if $Y$ is a $\mathbb R-$valued $\mathcal F^X_t-$measurable random variable, there exists a Borel function $f$ such that $$Y = f \big( (X_s)_{s\le t} \big).$$ My question is : is this still true in general if Y is $\overline{\mathcal F}^X_t-$measurable, where $\left(\overline{\mathcal F}^X_t\right)_{t\in [0,T]}$ is the completion of the filtration $(\mathcal F^X_t)_{t\in [0,T]}$ ?
Thank you very much for your time !