The spectral theorem and direct integrals

Question

I'm wondering if there are any good references that discusses the spectral theorem in terms of direct integrals? I suppose the statement would be something like this:

Let $N \in \mathcal{B}(H)$ be a normal operator. Then there exists a $\sigma$-finite Borel measure $\mu$ on $\sigma(N)$ and a measurable field of Hilbert spaces $\{ H(z): z \in \sigma(N) \}$ such that $H$ is isomorphic to $$ \int_{\sigma(N)}^\oplus H(z) \, d \mu(z) $$ and $N$ is unitarily equivalent to $$ \int_{\sigma(N)}^\oplus z \, d \mu(z). $$

Is this correct?

Answer 1

look for Quantum Theory for Mathematicians by Brian Hall it is an excellent on quantum theory , it contains the direct integral approach to spectral theorem