I'm looking for a proof that
Let $H$ be a separable Hilbert space and $N$ an unbounded normal operator on $H$. Then there is a finite measure space $(X,\scr{M},\mu)$, a unitary operator $U:H\to L^2(\mu)$, and a complex measurable function $f$ such that: (i) for $x\in H$, $x\in D(N)$ iff $f\,Ux\in L^2(\mu)$; (ii) for $\phi\in U(D(A))$, $UNU^{-1}\phi=f\phi$. In other words, $N$ becomes multiplication by $f$.
If $N$ is bounded, this can be found in A Course in Functional Analysis by G. B. Folland; however this book does not address unbounded operators. If $N$ is unbounded but self-adjoint, this can be found in Methods of Modern Mathematical Physics I: Functional Analysis by M. Reed & B. Simon; however this book does not seem to care about unbounded normal operators.
Thanks in advance!