TWO SUFFICIENT CONDITIONS FOR RECTIFIABLE MEASURES

We identify two sufficient conditions for locally finite Borel measures on R-n 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 mu on R-n satisfying the global regularity hypothesis lim (r down arrow 0) sup mu (B(x, r)) / r(m) < infinity at mu-a.e. x is an element of R-n 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 mu on R-n with lim r down arrow 0 mu (B(x, r)) / r - infinity at mu-a.e. x is an element of R-n is 1-rectifiable.