= Spectral measure of a normal operator
{wiki}
For a <normal operator> $A$ with <projection-valued measure> $E$ and vectors $f,g$, its scalar <spectral measure of a normal operator> is
$$
\mu_{f,g}(B)=\langle E(B)f,g\rangle.
$$
This is generally a complex measure, while $\mu_f=\mu_{f,f}$ is positive with total mass $\|f\|^2$. For every bounded Borel function $h$, the <Borel functional calculus for a normal operator> satisfies
$$
\langle h(A)f,g\rangle=\int h(z)\,d\mu_{f,g}(z).
$$
For an unbounded $h$, one must additionally have $f\in D(h(A))$, equivalently $\int|h|^2\,d\mu_f<\infty$. This domain condition makes the pairing and scalar integral well-defined; an arbitrary unbounded function cannot be inserted without checking it.
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