This is a proof from Rene Schilling's Brownian Motion that the completion of the $\sigma$-algebra generated by the BM $(B_t)_{t\ge 0}$ is admissible, i.e. the filtration at $s$ is independent of $B_t - B_s$ for all $s<t$.
The proof shows that $E(1_F \cdot (B_u-B_t)) = P(F)E(B_u - B_t)$ for all $F$ in the completion. How does this give independence?
What this seems to show is that a random variable $X$ is independent of a sigma-algebra $\mathscr{G}$ if $E(1_F \cdot X) = P(F) E(X)$ for all $F \in \mathscr{G}$, i.e. $E[X|\mathscr{G}] = E[X]$. But I know that this does not necessarily imply independence. So why is this proof complete?
