How "generalized eigenvalues" combine into producing the spectral measure?

Wouldn't Theorem 4.2 in here answer your question?

Theorem 4.2. (Generalized Eigenfunctions, by Mustafa Kesir) Let $\mathcal H$ be a Hilbert Space. Given a self-adjoint operator $A$ in $\mathcal H$ and a Hilbert–Schmidt rigging of $\mathcal H$, there exists a complete system of generalized eigenfunctions of $A$.