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Material Type: Assignment; Class: SPECIAL TOPICS; Subject: Mathematics; University: University of Washington - Seattle; Term: Unknown 1989;
Typology: Assignments
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Problem 1. Let μ be a Radon measure on Rn. Assume that for all a ∈ spt μ = Σ
c = lim sup
μ(B(a, 2 r)) μ(B(a, r))
Show that for μ a.e. a ∈ Rn, if ν ∈ T an(μ, a) then ν is a doubling measure, i.e. for each compact set K ⊂ Rn^ there exists a constant CK > 0 such that for x ∈ spt ν ∩ K, and r > 0
ν(B(x, 2 r)) ≤ CK ν((B(x, r)).
Note: If the general case is too technically involved you may assume that
sup a∈Σ
c(a) < ∞.
