Let $A\_1$ and $A\_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\varphi\_1,\varphi\_2\in L^\infty(\mu)$ and a unitary operator $U:E\longrightarrow L^2(\mu)$, such that each $A\_k$ is unitarily equivalent to multiplication by $\varphi\_k$, $k=1,2$. i.e.
$$UA\_kU^\*f=\varphi\_kf,\;\forall f\in E,\,k=1,2.$$


It is well known that if $E$ is **separable** then $\mu$ can be taken $\sigma$-finite.



> 
> Now if $E$ is assumed to be not necessarily separable, according to this [answer](https://mathoverflow.net/questions/294621/semifinite-measure-and-spectral-theorem), $\mu$ can be taken [localizable](https://www.math.wustl.edu/%7Enweaver/localizable.pdf).
> 
> 
> Does anyone have an exact reference?
> 
> 
> 


Thanks.



References for the Spectral Theorem ( Multiplication Operator Form)
You can see for example section 1.4 - spectral theorem II(1.47) - in the book "A course in abstract harmonic analysis" by "Gerald B. Folland".


