  In this paper I explore a nonstandard formulation of Hausdorff dimension. By
considering an adapted form of the counting measure formulation of Lebesgue
measure, I prove a nonstandard version of Frostman's lemma and show that
Hausdorff dimension can be computed through a counting argument rather than by
taking the infimum of a sum of certain covers. This formulation is then applied
to obtain a simple proof of the doubling of the dimension of certain sets under
a Brownian motion.
