Two sufficient conditions for rectifiable measures

Abstract

We identify two sufficient conditions for locally finite Borel measures on R to give full mass to a countable family of Lipschitz images of R. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L affine approximability for a locally finite Borel measure μ on R satisfying the global regularity hypothesis lim sup r↓0 μ(B(x, r))/r < ∞ at μ-a.e. x ∈ R to be m-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure μ on R with lim r↓0 μ(B(x, r))/r = ∞ at μ-a.e. x ∈ R