TWO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES

Abstract. We identify two sufficient conditions for locally finite Borel measures on Rn to give full mass to a countable family of Lipschitz images of Rm. The first condition, extending a prior result of Pajot, is a sufficient test in terms of Lp affine approximability for a locally finite Borel measure µ on Rn satisfying the global regularity hypothesis lim sup r↓0 µ(B(x, r))/rm < ∞ 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 lim r↓0 µ(B(x, r))/r = ∞ at µ-a.e. x ∈ Rn is 1-rectifiable. 1