APPROXIMATE TANGENTS, HARMONIC MEASURE, AND DOMAINS WITH RECTIFIABLE BOUNDARIES

Let Omega subset of Rn+1, n >= 1, be an open and connected set. Set T-n to be the set of points xi is an element of partial derivative Omega so that there exists an approximate tangent n-plane for partial derivative Omega at. and. O satisfies the weak lower Ahlfors-David n-regularity condition at xi. We first show that T-n can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting partial derivative(star)Omega be a subset of T-n where Omega satisfies an appropriate thickness condition, we prove that partial derivative(star)Omega can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Omega. As a corollary we obtain that if Omega has locally finite perimeter, partial derivative Omega is weakly lower Ahlfors-David n-regular, and the measure-theoretic boundary coincides with the topological boundary of Omega up to a set of H-n-measure zero, then partial derivative Omega can be covered, up to a set of H-n-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in Omega. This implies that in such domains, H-n vertical bar partial derivative Omega is absolutely continuous with respect to harmonic measure.