Canonical decomposition into pure point, absolutely continuous and singular continuous parts

For any finite measure nu, take its decomposition [A2]. Let P be the countable set of atoms, carrying the discrete part, and choose a Lebesgue-null Borel carrier N for the singular continuous part. The three disjoint Borel sets $S_{pp}=P$, $S_{sc}=N\setminus P$, $S_{ac}=\mathbb{R}\setminus (P\cup N)$ partition the line and carry their corresponding parts: both atomless parts vanish on P, the absolutely continuous part vanishes on P union N, and the discrete part vanishes off P. Thus $\nu {\mathrm{\mid }}_{S_r}={\nu }_r$ for each type r. More generally, for any disjoint carriers S_r of the three components, their union carries nu and the same restriction identity holds. If eta<<nu is finite positive, its restrictions to these carriers are of the respective types: they are dominated in the sense of null sets by nu_r, so inherit a countable carrier, Lebesgue absolute continuity, or a null carrier and zero singleton masses. They sum to eta, so uniqueness in [A2] implies eta has type r exactly when eta is carried by S_r. The zero measure has all three types, consistently with this assertion.

For every Borel B, [A3] gives $E_{x+y}(B)\le 2E_x(B)+2E_y(B)$ and $E_{cx}(B)=\mathrm{\mid }c{\mathrm{\mid }}^2{E}_x(B)$. Consequently each type set is linear: two countable carriers have countable union, two null carriers have null union, and zero masses on null sets or singletons pass through this inequality. If x_n of one type converge to x, the contraction inequality $\mathrm{\parallel }E(B)(x-x_n)\mathrm{\parallel }\le \mathrm{\parallel }x-x_n\mathrm{\parallel }$ shows E_x(B)=0 whenever all E_{x_n}(B)=0. For the discrete case choose countable carriers P_n and use their countable union P; then E_x(P^c)=0. For the singular case choose null Borel carriers N_n and use their null union N. For the atomless condition apply the same argument to each singleton; for absolute continuity apply it to each fixed Lebesgue-null Borel set. Hence all three subspaces are closed. The countable carrier choices and countable unions use the declared AC [A5].

Measures of different types are mutually singular: a discrete carrier is countable and both other types give it zero mass; a singular-continuous null carrier has zero absolutely continuous mass. Thus for x,y of different types there is a Borel S carrying E_x with E_y(S)=0. Since $\mathrm{\parallel }E(S^c)x{\mathrm{\parallel }}^2=E_x(S^c)=0$, one has E(S)x=x, while E(S)y=0. Self-adjointness of E(S) yields $⟨x,y⟩=⟨E(S)x,y⟩=⟨x,E(S)y⟩=0$.

For arbitrary x, apply step 1.1 to E_x and set x_r=E(S_r)x. The PVM identities give $E_{x_r}(B)=\mathrm{\parallel }E(B)E(S_r)x{\mathrm{\parallel }}^2=E_x(B\cap S_r)$, so x_r belongs to the indicated type. The partition gives $x=x_{pp}+x_{ac}+x_{sc}$. By step 2.1 this decomposition is orthogonal and unique. The component maps Q_r are linear by uniqueness, contractive by the Pythagorean identity for this finite orthogonal sum, and self-adjoint because $⟨Q_{r}x,y⟩=⟨x_r,y_r⟩=⟨x,Q_{r}y⟩$. They are orthogonal projections onto the closed type subspaces. As these subspaces were defined from E_x, they and the projections are canonical, independent of the carriers chosen for individual vectors.

For each Borel B the equality $E_{E(B)x}=E_x{\mathrm{\mid }}_B$ shows that E(B) preserves each type. Applying it to the unique decomposition in step 3.1 yields Q_r E(B)=E(B)Q_r. Therefore $E_{Q_{r}x}(B)=\mathrm{\parallel }{Q}_{r}E(B)x{\mathrm{\parallel }}^2\le E_x(B)$, so x in D(T) implies Q_rx in D(T) by [A4]. Commutation with every E(B) gives commutation with every simple integral and then every bounded integral by [A4]. Passing to the coordinate truncation limit gives TQ_rx=Q_rTx for x in D(T). Thus each type subspace reduces T, with its domain carried along.

In the separable clause let the supplied finite maximal measure mu have disjoint carriers B_r of its three components. For every x, E_x<<mu, so step 1.1 says x has type r exactly when E_x(B_r^c)=0. The latter is equivalent to E(B_r)x=x, since E(B_r^c)=I-E(B_r) and $E_x(B_r^c)=\mathrm{\parallel }E(B_r^c)x{\mathrm{\parallel }}^2$. Hence H_r=ran E(B_r). This proves the assertion for every supplied maximal mu, not just for a specially constructed one, and needs no circle-to-line transport.

On a nonzero type subspace K, E_K(B)=E(B)|_K is a regular PVM: its projection and strong countable-additivity properties restrict from E, and its scalar measures are the same regular E_x for x in K. The coordinate integral against E_K has domain K intersect D(T); bounded simple integrals and their limits agree with the restrictions of those for E, so its value is Tx. The converse spectral theorem in [A4] makes this restriction self-adjoint. On K={0}, self-adjointness is direct since its unique densely defined operator equals its adjoint. The scalar measures of each restriction have exactly the specified type by [A1].

If H={0}, every scalar measure is zero, the three subspaces are {0}, and all conclusions including the carrier formula hold directly. Vanishing components on a nonzero H also give zero subspaces by the same arguments, and no measure is divided by its mass. Nonseparability causes no difficulty in the preceding arguments, since only a single scalar measure or a sequence of vectors is used at a time. Full AC is used exactly as in [A5], including countable carrier choices for closedness.