I'll start by restating the Spectral Theorem for self-adjoint (unbounded) linear operators, in order to clarify the notation:
Let $H$ be a separable complex Hilbert space and $T:D(T)\subseteq H\rightarrow H$ a self-adjoint operator. Then, there exists a measurable space $(Y,\mu)$, with finite $\mu$, a unitary operator $U:H\rightarrow L^2(Y,d\mu)$ and a $\mu$-measurable function $q:Y\rightarrow \mathbb{R}$ such that:
i) $x\in D(T)\Leftrightarrow Ux\in D(M_q)$,
ii) $Tx=U^*M_qUx$.
Now, I want to show that $q$ can be chosen to be in $L^p(Y,d\mu)$, for any $p\in [1,\infty)$. I am looking at the proof of this Theorem but I'm kind of lost:
I know $q$ can be taken as $\frac 1m-i$, where $m$ is the function that comes from applying the Spectral Theorem for Bounded Normal operators to $(T-i)^{-1}$. Said $m$ is measurable and non-zero $\mu$-a.e. but this is as far as I got. What properties can I derive about $m$ in order to show that $q\in L^p(Y,d\mu)$ ? Am I missing some measure theory details perhaps?
Thanks in advance for your help, any hint is kindly appreciated.