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I see authors make statements such as: Suppose $W$ is Brownian motion realized on the canonical space $(C(\mathbb{R}_+; \mathbb{R}), \mathscr{B}(C(\mathbb{R}_+; \mathbb{R})))$. I'm quite confused about why this notion is introduced. What exactly does this mean? Is the following interpretation I give below correct?

If $P$ is the Wiener measure and $W_t(\omega) = \omega(t)$ for $\omega \in C(\mathbb{R}_+, \mathbb{R})$ and we consider the probability space $(C(\mathbb{R}_+; \mathbb{R}), \mathscr{B}(C(\mathbb{R}_+; \mathbb{R})), P)$ and filtration $\mathbb{F} = \{\sigma(W_s: s\leq t)\}_{t \ge 0}$, then $W$ is Brownian motion under $P$.

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    $\begingroup$ Yes, that is exactly right. $\endgroup$ – Jason Apr 7 at 18:22

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