Doubling measures with doubling continuous part

Abstract

We prove that every compact subset of R d \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 E of  R d \mathbb R^d without isolated points and for every doubling measure μ \mu on E E there is a countable set F F with E ∩ F = ∅ E\cap F=\emptyset and a doubling measure ν \nu on E ∪ F E\cup F such that ν | E = μ \nu |_E=\mu . This shows that there are many doubling measures whose continuous part is doubling.