# Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result :

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it has an area ? If yes, has anybody measured or calculated its "size" (Hausdorff measure) ? Thanks.

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–  J. M. Aug 31 '10 at 8:26
A planar set can esily have zero measure and Hausdorff dimension 2. For example, delete every point having a coordinate with normal binary expansion from the unit square. –  John Bentin Aug 31 '10 at 12:33