The exact Hausdorff measure of the sample path for planar Brownian motion

It has long been known (see Levy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to t α is infinite for 0 C ω = C (1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ( t ) such that, with probability 1, for suitable positive constants c 1 , c 2 . The corresponding problem for paths in k -space ( k ≥ 3) has been solved. In this case, if φ 1 ( t ) = t 2 log log t −1 , Levy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ 1 ( t ) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ 2 ( t ) = t 2 log t −1 , and at that time we thought that (1) might be true with ψ( t ) = φ 2 ( t ). Recently Ray (5) has obtained the lower bound in (1)with The purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.