 Spectral measure of a normal operator

For a normal operator $A$ with projection-valued measure $E$ and vectors $f,g$, its scalar spectral measure of a normal operator isThis is generally a complex measure, while ${\mu }_f={\mu }_{f,f}$ is positive with total mass $\Vert f{\Vert }^2$. For every bounded Borel function $h$, the Borel functional calculus for a normal operator satisfiesFor 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.