DOUBLING MEASURES WITH DOUBLING CONTINUOUS PART

We prove that every compact subset of 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 R-d without isolated points and for every doubling measure mu on E there is a countable set F with E boolean AND F = empty set and a doubling measure nu on E boolean OR F such that nu vertical bar(E) = mu. This shows that there are many doubling measures whose continuous part is doubling.