The Hausdorff α-dimensional measure of Brownian paths in n-space

Abstract

Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3. However, if we apply the method indicated by Hausdorff (4) for constructing a linear set having measure 1 with respect to a given measure function h(x), the Cantor-type set we obtain when h(x) = xα log log 1/x, where 0 < α < 1, is easily seen to have zero α-capacity but infinite α-measure; and similar methods apply for other values of α. Thus the result mentioned above does not imply that Brownian paths in n-space (n ≥ 3) have zero 2-measure. The other relevant result is due to Lévy (6), who showed that Brownian paths in the plane have zero Lebesgue measure (and therefore zero Hausdorff 2-measure) with probability 1. However, his method of proof cannot be extended to deal with Brownian paths in n-space.