Let $(\Omega, \mathscr{F}, P)$ be a probability space and $T>0.$ Also, let $\{B_t; 0 \leq t \leq T\}$ be a Brownian motion that generates the filtration $\{\mathscr{F}_t^W; 0 \leq t \leq T\}.$ I have read in a couple of math finance books: "assume $\mathscr{F}_T^W = \mathscr{F}...".$ How can we justify this assumption?
In order to define a B.M. we need a probability space, right? So we start with a probability space with some sigma-algebra defined on Ω and then we define a B.M. that generates the filtration. And then we replace the original sigma-algebra by $\mathscr{F}_T$? How can we do that? It seems to me like "the chicken or the egg".