  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
$\frac{1}{n} K_n(x,x) d\mu(x)$. By combining this with Mate--Nevai and Totik
upper bounds on $n\lambda_n(x)$, we prove some general results on $\int_I
\frac{1}{n} K_n(x,x) d\mu_s\to 0$ for the singular part of $d\mu$ and $\int_I
|\rho_E(x) - \frac{w(x)}{n} K_n(x,x)| dx\to 0$, where $\rho_E$ is the density
of the equilibrium measure and $w(x)$ the density of $d\mu$.
