Let $p\in [0,1].$ I am interested in showing that there exist sets $A,B\subset \mathbb{R}$ of Hausdorff dimension $p$ such that the $p$-dimensional Hausdorff measures $H_p(A)=\infty$ and $H_p(B)=0$.
I'm pretty much stuck on how to proceed, so any help in the right direction will be appreciated.
For $0 < p < 1$ you can modify the construction of the Cantor set to get a bounded set of finite nonzero $H_p$ measure. To get a set of dimension $p$ and $H_p(A) = 0$, take the union of a sequence of such sets with Hausdorff dimension approaching $p$ from below. To get a set of dimension $p$ and $H_p(A) = \infty$, take the union of a sequence of sets with finite $p$-dimensional Hausdorff measure going to $\infty$ (e.g. scaled versions of one such set).