Problem
Given a Hilbert space $\mathcal{H}$.
Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$
Denote its probability measures by: $$\nu_\varphi(A):=\|E(A)\varphi\|^2$$
Introduce the pure-point space: $$\mathcal{H}_0(E):=\{\varphi:\exists\#\Lambda_0\leq\aleph_0:\nu_\varphi(\Lambda_0)=\nu_\varphi(\Omega)\}$$
Construct its normal operator: $$\varphi\in\mathcal{D}(N):\quad\langle N\varphi,\chi\rangle=\int_\mathbb{C}\lambda\mathrm{d}\langle E(\lambda)\varphi,\chi\rangle\quad(\chi\in\mathcal{H})$$
Regard its eigenspace: $$\mathcal{E}_\lambda=\{\varphi:N\varphi=\lambda\varphi\}:\quad\mathcal{E}(N):=\cup_{\lambda}\mathcal{E}_\lambda$$
Then one has: $$\mathcal{H}_0(E)=\overline{\langle\mathcal{E}(N)\rangle}$$
How to prove this?
Reference
This thread is related to: Spectral Spaces (I)