We show that two-dimensional super-Brownian motion is a multiple of the $h$-Hausdorff measure on its closed support, where $h(r) = r^2 \log^+(1/r)\log^+ \log^+ \log^+(1/r)$. This complements known results in dimensions greater than 2.
Jean-Francois Le Gall. Edwin A. Perkins. "The Hausdorff Measure of the Support of Two-Dimensional Super-Brownian Motion." Ann. Probab. 23 (4) 1719 - 1747, October, 1995. https://doi.org/10.1214/aop/1176987800