Two examples on atomic doubling measures

It is shown that there is a compact set X ⊂ [0, 1] with dimH X = 1 on which all doubling measures are purely atomic. It is also shown that there is a compact set X ⊂ [0, 1] with a dense set of isolated points and dimH X = 0 on which no doubling measures are purely atomic. © 2006 Elsevier Inc. All rights reserved