The following is taken from a proof of Lévy's construction of Brownian motion in a book by Peter Mörters and Yuval Peres.
$\mathcal {D_n } := \{\frac {k } {2^n } :1 \le k \le 2^n \} $, the set of dyadic points in $[0,1]$.
I'm not sure how the claim in $(2)$ is defined and thus have a hard time reading the paragraph that follows.
My first guess was that it mean that each element of the first vector is independent of the second vector seen as a collection of random variables. This seems to be implicated by the last sentence.
But then to conclude that $\frac {B(d-2^{-n }) + B(d+2^{-n })} {2 } + \frac {Z_d } {2^{(n+1)/2 } } $ is independent of $\{Z_t:t \in \mathcal {D-D_n } \} $ I believe we also need $B(d+2^{-n }),B(d+2^{-n })$ and $Z_d $ to be mutually independen. Is this coorect?
Many thanks in advance!