Bulk universality holds in measure for compactly supported measures

Let mu be a measure with compact support, with orthonormal polynomials {p(n)} and associated reproducing kernels {K-n}. We show that bulk universality holds in measure in {xi : mu' (xi) > 0}. More precisely, given epsilon, r > 0, the linear Lebesgue measure of the set {xi : mu' (xi) > 0} and for which sup(vertical bar u vertical bar,vertical bar v vertical bar <= r) vertical bar K-n(xi + u/(K) over tilde (n)(xi,xi) + v/(K) over tilde (n)(xi,xi))/K-n(xi,xi) - sin pi(u - v)/pi(u - v)vertical bar >= epsilon approaches 0 as n -> infinity. There are no local or global regularity conditions on the measure mu.