Usually, a filtered probability space is given by $(\Omega, \mathcal F, \mathbb P, \{\mathcal F_t\}_{t\ge 0})$ on which $W$ is a Brownian motion. Sometimes, it may be desirable to work with a specific sample space $\Omega$ rather than the above abstract one. That gives a setup of canonical space, roughly speaking, $\Omega = C([0, \infty), \mathbb R^{d})$ and $\mathcal F_t^o = \sigma(W_s: 0 \le s \le t)$ is a natural filtration, and $\mathcal F_t$ is its complete augmentation. However, it may be still desirable to have more concrete $\mathcal F_t$ induced by metric. Therefore, I made the following statement, which is not seen from any book.
[Q.] Is there any technical flaw on the probability space setup in this below?
Let the sample space be $\Omega = C([0, \infty), \mathbb R^{d})$. Then, we can define various supnorms up to a given time intervals: $$\|\omega\| = \sup_{0\le s < \infty} |\omega(s)|, \ \|\omega\|_{t} = \sup_{0\le s \le t} |\omega(s)|. $$ We denote by $\mathcal F$ its Borel $\sigma$-algebra induced by $\|\cdot \|$, and by $\mathcal F_{t}^{o}$ its Borel $\sigma$-algebra induced by $\|\cdot\|_{t}$ for each $t\in [0, \infty)$. Let $\mathbb W$ be the Wiener measure on $(\Omega, \mathcal F)$ and $\{\mathcal F_{t}: t\ge 0\}$ is its complete augmented filtration, i.e. $\mathcal F_{t}$ is the smallest $\sigma$-algebra containing $ \bigcap_{s > t} \mathcal F_{t}^{o}$ and all $\mathbb W$-null sets. Then, the random process $W: \Omega \times [0, \infty) \mapsto \mathbb R^{d}$ defined by $W(\omega, t) = \omega(t)$ is $\mathbb W$-Brownian motion.