Doubling measures on Whitney modification sets

A Whitney modification set in R is a set of the form E ∪ F(W), where E is a nonempty closed set in R, W is a Whitney decomposition of R \ E, and F(W) consists of centers of intervals in W. We prove that a measure on a Whitney modification set is doubling if and only if it is the Whitney modification of a doubling measure on R. © 2007 Elsevier Inc. All rights reserved.