Problem
Given a Hilbert space $\mathcal{H}$.
Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$
By the previous threads: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad N=Z\left(\sqrt{1-Z^*Z}\right)^{-1}$$
Especially one had: $$Z=\int\lambda\mathrm{d}F:\quad F(\overline{\mathbb{D}})=1\quad F(\mathbb{S})=0$$
Define the function: $$\eta\in\mathcal{B}(\mathbb{D}):\quad\eta(\lambda):=\frac{\lambda}{\sqrt{1-|\lambda|^2}}$$
Construct as measure: $$E(A):=F_\eta(A):=F(\eta^{-1}A)$$
Then one obtains: $$N=\int\lambda\mathrm{d}F_\eta(\lambda)=:\int\lambda\mathrm{d}E(\lambda)$$
How can I prove this?
Reference
This builds up on: Tranform, Retransform