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Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$
Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book.
Suppose ...