Doubling measures with doubling continuous part

We prove that every compact subset of $\mathbb{R}^d$ of positive Lebesgue measure carries a doubling measure which is not purely atomic. Also, we prove that for every compact and nowhere dense subset $E$ of  $\mathbb{R}^d$ without isolated points and for every doubling measure $\mu$ on $E$ there is a countable set $F$ with $E\cap F=\emptyset$ and a doubling measure $\nu$ on $E\cup F$ such that $\nu\vert _E=\mu$. This shows that there are many doubling measures whose continuous part is doubling.