Can You Calculate the Hausdorff Measure of Fractals Accurately?

[nomadreid]

I am confused about the calculation of Hausdorff measure. On one side, the Hausdorff dimension is calculated as the point where the Hausdorff measure of the figure jumps from infinity down to zero (formally, in terms of limits of the inf and sup of the measures of the covers, etc.) But that just tells me what the measure of the figure is above and below that point, not at the point itself. So, can one actually calculate the Hausdorff measure of fractals? All the sites I read only talk about calculating the Hausdorff dimension.

For example, the Hausdorff measure is proportional to the Lebesgue measure for positive integer dimensions. Does this proportionality extend to positive reals? It would seem to, given that the proportionality is given in terms of the gamma function.

If so, then since the Lebesgue measure of the Cantor set is zero, the Hausdorff measure should also be, whereupon the scaling property would seem to lose all meaning.

 

[pat_connell]

The answer to your question is yes, you can actually calculate the Hausdorff measure of fractals. The Hausdorff measure of a fractal is simply the measure of the set of points on the fractal at a given dimension. For example, the Hausdorff measure of the Cantor set at dimension 0 is 1, since there is one point at that dimension. At dimension 1/2, the Hausdorff measure is 0, since there are no points at that dimension. The same holds true for any fractal: the Hausdorff measure is simply the number of points at a given dimension.As for the proportionality between the Hausdorff measure and the Lebesgue measure, it does extend to positive reals, as you suspected. The scaling factor between the two measures is given by the gamma function. This means that if the Lebesgue measure of a set is zero, then so is its Hausdorff measure.