On the Hausdorff Measure of Brownian Paths in the Plane

1. Introduction. 4 will denote the space of all plane paths w, so that w is a short way of denoting the curve z(t, (o) = {x(t, w), y(t, (o)} (0 < t < + oc). We assume that there is a probability measure ,u defined on a Borel field 9 of (measurable) subsets of S2, so that the system (Q, 3~ 7_ " u) forms a mathematical model for Brownian paths in the plane. [For details of the definition of ,u, see for example (9) .] For 0< a< b<+ oo, w E S2, let L(a, b ; (o) be the plane set of points z(t, (o) for a < t < b. Then with probability 1, L(a, b ; w) is a continuous curve in the plane. The object of the present note is to consider the measure of this point set L(a, b ; (o). The first step in this direction is due to Levy (7) who proved that, with probability 1, the Lebesgue plane measure of L(0, oo ; (o) is zero. In (9), one of us considered the fractional dimension measure, that is the measure with respect to the function x8 (0 < s < 2). It was proved that, again with probability 1, the measure with respect to each xs (0 < s < 2), is infinite : that is, the path has dimension 2 in the sense of Besicovitch. In (8), Levy improved his zero Lebesgue measure result by proving that the measure of L(0, 1 ; w) with respect to the function x 2 loglog 1/x is finite with probability 1. In fact Levy proves this result for Brownian paths in n-dimensional Euclidean space (n > 2), but states that he does not expect his result to be best possible for paths in the plane. He conjectured that in the plane case the measure of L(0, 1 ; (0) is finite with respect to the function x 2 log 1/x. O (x) is called a measure function if there is a S > 0 such that O(x) is monotonic increasing and continuous for 0 < x < S and lim O(x) = 0. For a set of points E in a , 0 + Euclidean space the Hausdorif measure of E with respect to O(x), first defined in (4), is denoted by 0-m(E). Put hx (x) = x2 (log 1/x)x (a > …