Pure point, absolutely continuous and singular continuous spectral subspaces

Assume the Axiom of Choice. Let $T$ be a self-adjoint operator on $H$. If $H\mathrm{\ne }\{0\}$, let $E$ be its spectral projection valued measure on $\mathbb{R}$ from Spectral theorem for unbounded self-adjoint operators (PVM form); if $H=\{0\}$, let $E(B)=0$ for every Borel $B$, the unique PVM on the zero space. In either case put $E_x(B)=⟨E(B)x,x⟩$ (Integral of a measurable function against a projection-valued measure). Call a finite Borel measure on $\mathbb{R}$ purely atomic (equivalently, discrete) when it is concentrated on a countable subset of $\mathbb{R}$. Then $H_{\mathrm{p}\mathrm{p}}=\{x\in H:E_x\text{ is purely atomic}\},$ $H_{\mathrm{a}\mathrm{c}}=\{x\in H:E_x\ll \lambda \text{ (Lebesgue measure)}\},$ $H_{\mathrm{s}\mathrm{c}}=\{x\in H:E_x\text{ is atomless and singular with respect to }\lambda \},$ where atoms are as in An atom of a measure on $\mathbb{R}$, absolute continuity is that of Absolute continuity of a signed or complex measure with respect to a positive measure and singularity that of Mutual singularity for signed or complex measures.

Well-definedness. Every finite Borel measure on $\mathbb{R}$ has a unique decomposition into a discrete (hence, by the convention above, purely atomic), an absolutely continuous and an atomless singular part (Every finite Borel measure on R has a unique absolutely continuous, discrete, and singular-continuous decomposition), and the three classes of nonzero measures are mutually exclusive; hence each $x$ either satisfies one of the three defining conditions or none, and the zero vector lies in all three subspaces. Whether the three subspaces do cover $H$ and are closed is not part of this definition and is the content of the canonical spectral type decomposition theorem below.

Conventions. For each type one writes ${\sigma }_{\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}}(T)=\sigma (T{\mathrm{\mid }}_{H_{\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}}})$ for the restriction of $T$ to the closed invariant subspace $H_{\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}}$ constructed in the canonical spectral type decomposition theorem below; these restrictions are self-adjoint there. The point spectrum is not ${\sigma }_{\mathrm{p}\mathrm{p}}$: ${\sigma }_{\mathrm{p}\mathrm{p}}(T)$ is the closure of the set of eigenvalues of $T$, and the three sets ${\sigma }_{\mathrm{p}\mathrm{p}},{\sigma }_{\mathrm{a}\mathrm{c}},{\sigma }_{\mathrm{s}\mathrm{c}}$ may overlap, so they do not partition $\sigma (T)$.