Weak convergence of scalar spectral measures

Scalar spectral measures ${\mu }_{v,w}^{(n)}$ converge weakly to ${\mu }_{v,w}$ when $\int fd{\mu }_{v,w}^{(n)}\to \int fd{\mu }_{v,w}$ for every bounded continuous function $f$. Uniform second-moment bounds supply the tightness needed to pass from polynomial tests to all such $f$.