We prove a general result on equality of the weak limits of the zero counting measure, $d\nu_n$ , of orthogonal polynomials (defined by a measure $d\mu$ ) and $({1}/{n}) K_n (x,x) d\mu(x)$ . By combining this with the asymptotic upper bounds of Máté and Nevai [16] and Totik [33] on $n\lambda_n(x)$ , we prove some general results on $\int_I ({1}/{n}) K_n(x,x) d\mu_\rm{s}\to 0$ for the singular part of $d\mu$ and $\int_I \vert\rho_E(x) - ({w(x)}/{n}) K_n(x,x)\vert dx\to 0$ , where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$