Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes over all such probability measures. If A is strongly -rectifiable, then μ ^s converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below. This research was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0808093.