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