Spectral measure of a normal operator

For vectors $f,g$, the scalar spectral measure is ${\mu }_{f,g}(S)=⟨E(S)f,g⟩$. Functional calculus satisfies $⟨h(A)f,g⟩=\int hd{\mu }_{f,g}$.

Stone's formula recovers spectral projections of a self-adjoint operator from the jump of its resolvent across the real axis, with half weight at interval endpoints.