Math 582
Homework - Part 3
Due March 17
Problem 4. Let µbe a Radon measure on Rn. Assume that for a∈spt µ= Σ
(1) 1 ≤lim sup µ(B(a, 2r))
µ(B(a, r)) <∞.
1. Show that for τ≥1 and a∈Σ
1≤lim sup µ(B(a, τ r))
µ(B(a, r)) <∞.
2. Prove that if there exit κ > 1 and R > 0 such that for r∈(0, R) and all a∈Σ
(2) µ(B(a, 2r))
µ(B(a, r)) ≤κ
then for all ν∈T an(µ, a) such that
ν= lim
i→∞
(µ(B(a, ri)))−1Ta,ri#µ
x∈spt νif and only if there exists a sequence xi∈Ta,ri(Σ) such that xi→x.
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