# Inequality for Hausdorff Measure

I am reading Falconer's book "Fractal Geometry" and he mentions that we can assume that "$0<\mathcal{H}^{s}(F)<\infty$" where $\mathcal{H}^{s}$ is the $s$-dimensional Hausdorff measure. I get the feeling the he means that we can only do this in certain situations.

He uses it (along with the scaling property of the Hausdorff measure) in his heuristic calculation of the Hausdorff dimension of the Middle Third Cantor set as well as other self-similar sets.

I was just wondering if someone could tell me if and when we are allowed to assume that $s=\dim_{H}(F)$ and so $0<\mathcal{H}^{s}(F)<\infty$.

• I cannot find where this is stated. Can you give a more exact reference? To what page of which edition are you referring? – Xander Henderson Aug 14 '17 at 16:11
• @XanderHenderson In the second edition it is at the top of page 35, second line down. On this pdf: dm.uba.ar/materias/optativas/geometria_fractal/2006/1/Fractales/…, it is at the bottom of page 26 in the Heuristic Calculation bit. – user458984 Aug 16 '17 at 11:08

Assuming that at the critical value $s = \dim_H F$ we have $0 < \mathcal{H}^s(F) < \infty$ (a big assumption, but one that can be justified) we may divide by $\mathcal{H}^s(F)$ to get $2(\frac{1}{3})^2$ or $s = \log(2)/\log(3)$. [original emphasis]
Thus I think that the way to understand this passage is to read Falconer as saying "If this division is justified, then we may conclude that $s=\log(2)/\log(3)$." At the time that Falconer introduces this assumption, he has not yet justified it (though he knows that it can be justified and, indeed, justifies it in the next subsection).