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Hausdorff measure
The image of Brownian motion in \({\mathbf{R}}^2\) almost surely has Hausdorff dimension 2 and 2-dimensional Hausdorff measure 0.
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The image of Brownian motion in \({\mathbf{R}}^2\) almost surely has Hausdorff dimension 2 and 2-dimensional Hausdorff measure 0.