It is well-known, and well-documented, that the Hausdorff dimension of the graph of regular $1$-dimensional Brownian motion is $3/2$ (almost surely). See for example Theorem 4.29 in "Brownian Motion" by Mörters and Perez. Suprisingly, after looking through as much literature as I can, I have not seen addressed the question of the actual value of the Hausdorff $3/2$-measure on the graph of Brownian motion, say, on the interval $[0,1]$, and in particular whether this is zero, positive, or infinite. Is this known?
Two possibly useful pieces of evidence:
-By Remark 4.25 of Mörters-Perez, the Hausdorff $1/2$-measure on the zero set of Brownian motion vanishes. Maybe this implies, by integrating over the level sets, that the Hausdorff $3/2$-measure of the graph is also zero.
-On the other hand, Brownian motion fails to be $1/2$-Hölder continuous, which seems to leave the door open for the graph to have infinite Hausdorff $3/2$-measure.
