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.