Let $\Omega$ be the space of continuous functions $\omega: [0,T]\to \mathbb{R}^{d}$, $\mathcal{F}=\mathcal{B}(C[0,T))$ and $\mathbb{P}$ be the Wiener measure. Therefore the coordinate processes $W$ defined by $W_t(\omega)=\omega_t$ is a Brownian motion. Let $(\mathcal{F}_t)_{0\leq t<\infty}$ by the natural filtration generated by $W$. Is the following conclusion true?
For $0<t<s$ and a non-null set $A\in \mathcal{F}_s$, we can always find a non-null set $B$ satisfying that $B\subset A$ and $B\in\mathcal{F}_t$?