Example
Almost surely, a sample path X of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure μ2(X) is zero. The exact dimension function h is given by the logarithmic correction
I.e., with probability one, 0 < μh(X) < +∞ for a Brownian path X in R2. For Brownian motion in Euclidean n-space Rn with n ≥ 3, the exact dimension function is
Read more about this topic: Dimension Function
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