Quantitative Reifenberg theorem for measures

We study generalizations of Reifenberg’s Theorem for measures in Rn under assumptions on the Jones’ β-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which hold for general measures without density assumptions, give effective measure bounds on μ away from a closed k-rectifiable set with bounded Hausdorff measure. We show examples to see the sharpness of our results. Under further density assumptions one can translate this into a global measure bound and k-rectifiable structure for μ. Applications include quantitative Reifenberg theorems on sets and discrete measures, as well as upper Ahlfor’s regularity estimates on measures which satisfy β-number estimates on all scales.