Scalar spectral measure (${\mu }_{v,w}$)

For a projection-valued measure $E$ and vectors $v,w$, the scalar spectral measure is the complex measure ${\mu }_{v,w}(B)=⟨E(B)v,w⟩$. The positive measure ${\mu }_v={\mu }_{v,v}$ has total mass $\Vert v{\Vert }^2$.

A sequence of scalar spectral measures ${\mu }_{v,w}^{(n)}$ converges weakly to ${\mu }_{v,w}$ if its integrals against every bounded continuous function converge. A concrete sufficient setting is that all underlying normal-operator spectra lie in one fixed compact set $K\subset \mathbb{C}$ andThe Stone-Weierstrass theorem approximates any continuous function on $K$ uniformly by polynomials in $z,\overline{z}$. Also $\Vert {\mu }_{v,w}^{(n)}{\Vert }_{\mathrm{TV}}\le \Vert v\Vert \Vert w\Vert$: apply Cauchy-Schwarz to the disjoint spectral projections in any finite partition. These uniform total-variation bounds control the approximation errors, so the mixed polynomial tests imply all continuous tests.

On a noncompact real spectrum, a uniform second-moment bound gives uniform tightness for positive scalar spectral measures, but does not alone justify the polynomial approximation step. High-degree polynomial tails require additional uniform integrability, and identifying a limit from all moments needs an additional hypothesis ensuring that these determine the measure. Merely asserting tightness does not supply either missing condition.