Weak convergence of CD kernels and applications

We prove a general result on equality of the weak limits of the zero counting measure, dνndνn, of orthogonal polynomials (defined by a measure dμ) and (1/n)K_n(x,x)dμ(x). By combining this with the asymptotic upper bounds of Máté and Nevai [16] and Totik [33] on nλ_n(x), we prove some general results on ∫_I (1/n)K_n(x,x)d μ_s → 0 for the singular part of dμ and ∫_ I ∣∣ρ_E(x) − (w(x)/n)K_n(x,x)∣∣dx → 0, where ρ_E is the density of the equilibrium measure and w(x)x) the density of d μ.