On distribution of local dimensions of doubling measures on Euclidean space

We prove that for every nonnegative, increasing and right continuous function g on [0,n] with g(n)=1 and g(ϵ)=0 for some ϵ∈(0,n) there exists a doubling probability measure μ on [0,1]n such that the distribution Gμ of the lower local dimension of μ is g.