Two sufficient conditions for rectifiable measures

We identify two sufficient conditions for locally finite Borel measures on ℝⁿ to give full mass to a countable family of Lipschitz images of ℝ^m. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L^p affine approximability for a locally finite Borel measure μ on ℝⁿ satisfying the global regularity hypothesis μ(B(x, r))/r^m < ∞ at μ-a.e. x ∈ ℝⁿ 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 ℝⁿ with μ(B(x, r))/r = ∞ at μ-a.e. x ∈ ℝⁿ is 1-rectifiable.