  In this paper by calculating carefully the capacities (defined by high order
Sobolev norms on the Wiener space) for some functions of Brownian motion, we
show that the dyadic approximations of the sample paths of the Brownian motion
converge in the $p$-variation distance to the Brownian motion except for a slim
set (i.e. except for a zero subset with respect to the capacity on the Wiener
space of any order). This presents a way for studying quasi-sure properties of
Wiener functionals by means of the rough path analysis.
