$\textbf{Thm}$Let $A$ be an unbounded self-adjoint operator on a separable Hilbert space $H$ with domain $D(A)$. There there exists a measure sapce $(M,\mu)$ with $\mu$ a finite measure, a unitary operator $U:H\rightarrow L^2(M,\mu)$, and a real-valued function $f$ on $M$ which is finite a.e. so that $$\psi\in D(A) \text{ iff } f(.)(U\psi)(.)\in L^2(M,d\mu)$$ $$\text{If } \varphi \in U[D(A)], \text{then }(UAU^{-1})(m)=f(m)\varphi(m)$$
$\textbf{Example.}$ Let $A$ be an unbounded, self-adjoint operator on Hilbert space $H$. Let us pass to a spectral representation of $A$, so that $A$ is multiplication by $x$ on $\bigoplus_{n=1}^NL^2(\mathbb{R},\mu_n)$. Let $$Q(q)=\{(\psi_n(x))_{n=1}^N|\sum_{n=1}^N\int|x||\psi_n(x)|^2d\mu_n<\infty\}$$ For $\psi, \phi \in Q(q),$ $$q(\phi,\psi)=\sum_{n=1}^N\int x\overline{\phi_n(x)}\psi(x)du_n.$$
These two are material from Reed&Simon's book. I am having trouble understanding the line "Let us pass to a spectral representation of $A$, so that $A$ is multiplication by $x$ on $\bigoplus_{n=1}^NL^2(\mathbb{R},\mu_n)$." I dont know how to apply the theorem to get the multiplication by $x$ on $\bigoplus_{n=1}^NL^2(\mathbb{R},\mu_n)$. Anyone could possibly help me with that?