# A set with infinity Hausdorff measure, but Hausdorff dimension $\frac{\log2}{\log3}$

I am going through the text 'Fractal Geometry: Mathematical Foundations and Applications' and came the following exercise:

Give a set $$B\subset\mathbb{R}$$ that has Hausdorff dimension $$s=\frac{\log2}{\log3}$$ but has $$H^s(B)=\infty$$.

My thoughts:

Now, when I see $$\frac{\log2}{\log3}$$ the first thing that comes to mind is the cantor set $$C$$, but the $$H^s(C)$$ is finite. I feel as though I am missing a lemma or theorem. Is there some relation of the Hausdorff dimension of a set constructed by countable unions?

• If the union is nice enough (e.g. not two of the sets are nearer than a given $\delta > 0$), the Hausdorff dimension of the union is the supremum of the dimensions of the parts. Also note that Hausdorff measures are translation-invariant. – Daniel Fischer May 30 at 13:33
• So ive I took a countable number of cantor sets, some $\delta>0$ apart, then the Hausdorff dimension would still be $s$ but, the Hausdorff measure $\infty$? – kam May 30 at 13:35
• If your starting set has $H^s(C) > 0$. In this situation, the positive distance isn't needed (that's just a condition that works in every metric space), it suffices that there's no overlap. – Daniel Fischer May 30 at 13:43
• math.stackexchange.com/questions/3686906/… looks similar – Claude May 30 at 15:10
• math.stackexchange.com/questions/3695680 – Xander Henderson Jun 1 at 12:45