Multiplicity model of a projection-valued measure over a standard Borel base

Existence of a faithful measure: for a dense sequence $({\xi }_j)$ with ${\xi }_j\mathrm{\ne }0$ put ${\mu }_0={\sum }_j{2}^{-j}\mathrm{\parallel }{\xi }_j{\mathrm{\parallel }}^{-2}{E}_{{\xi }_j,{\xi }_j}$. This is a finite Borel measure, and ${\mu }_0(E)=0$ forces $E_{{\xi }_j,{\xi }_j}(E)=⟨P(E){\xi }_j,{\xi }_j⟩=0$ for every $j$, so $P(E){\xi }_j=0$ for every $j$; density of $({\xi }_j)$ and boundedness of $P(E)$ give $P(E)=0$; the converse is immediate. Hence ${\mu }_0$ is $P$-faithful. If ${\mu }_1,{\mu }_2$ are $P$-faithful, then ${\mu }_1(E)=0\iff P(E)=0\iff {\mu }_2(E)=0$, so they are mutually absolutely continuous.

Let $c:X\to [0,1]$ be a bimeasurable injection onto the Borel set $Y=c(X)$ by [F2], and put $S={\int }_{X}cdP$, a bounded self-adjoint operator by [F3]. Then $Q(D):=P(c^{-1}(D))$ for Borel $D\subseteq [0,1]$ is a projection-valued measure on $[0,1]$, because $D\mapsto c^{-1}(D)$ preserves the Boolean operations: $Q(\mathrm{\varnothing })=0$, $Q([0,1])=P(X)=I$, $Q(D_1)Q(D_2)=P(c^{-1}(D_1)\cap c^{-1}(D_2))=Q(D_1\cap D_2)$, and countable additivity transfers. Its coordinate integral is $S$: by [F1] and change of variables for the PVM, ${\int }_{[0,1]}zdQ(z)={\int }_{[0,1]}zdP(c^{-1}(z))={\int }_{X}c(x)dP(x)=S$. For $z\mathrm{\notin }[0,1]$, the bounded integral of $(z-t{)}^{-1}$ against $Q$ is a two-sided inverse of $zI-S$ by [F1], so $\sigma (S)\subseteq [0,1]$. Each scalar measure of $Q$ is a finite Borel measure on the compact metric space $[0,1]$, hence regular by [F6], so $Q$ is a regular PVM.

Spectral identification: by the uniqueness clause of [F3] applied to the regular PVM $Q$ on $\mathrm{\Lambda }=[0,1]$, one has $Q([0,1]\setminus \sigma (S))=0$ and $Q(D)=E_S(D)$ for every Borel $D\subseteq \sigma (S)$. Extend $E_S$ by zero outside $\sigma (S)$ when writing it on $[0,1]$. Consequently, since $c^{-1}(c(B))=B$, $P(B)=Q(c(B))=E_S(c(B))(B\subseteq X\text{ Borel}),$ with $c(B)$ Borel by bimeasurability; in particular $E_S$ is carried by $Y\cap \sigma (S)$ because $E_S(c(X))=P(X)=I$. Set $\mathcal{A}:=W^{\ast }(S)$; it is abelian because polynomials in the self-adjoint $S$ commute and commutation with a fixed bounded operator is WOT closed, so their WOT closure still commutes pairwise. Thus $S$ is a prescribed self-adjoint generator to which [F4] applies.

Apply the spectral multiplicity model of [F4] to the pair $(\mathcal{A},S)$: there are a finite regular Borel measure $\nu$ on $K=\sigma (S)$, a Borel function $m:K\to \{1,2,\dots \}\cup \{\mathrm{\infty }\}$ and a unitary $U:H\to {\int }_K^{\oplus }{\mathbb{C}}^{m(t)}d\nu (t)$ with $US{U}^{-1}=M_t$ and $U\mathcal{A}{U}^{-1}=\{M_f\}$. Since $U{\mathbf{1}}_D(S)U^{-1}={\mathbf{1}}_D(M_t)=M_{{\mathbf{1}}_D}$, the operator $U{E}_S(D)U^{-1}$ is the multiplication by ${\mathbf{1}}_D$, so $\nu$ is $E_S$-faithful: $\nu (D)=0$ iff $M_{{\mathbf{1}}_D}=0$ iff $U{E}_S(D)U^{-1}=0$ iff $E_S(D)=0$, the middle equivalence using that every fibre is nonzero so that a multiplication operator is zero exactly when its symbol vanishes a.e.

The pushforward ${\nu }_0:=c_{\ast }\mu$ restricted to $Y$ is likewise $E_S$-faithful: for Borel $D\subseteq [0,1]$ one has ${\nu }_0(D)=\mu (c^{-1}D)=0$ iff $P(c^{-1}D)=E_S(c(c^{-1}D))=E_S(D\cap Y)=E_S(D)$, using $P$-faithfulness of $\mu$ and $E_S$ being carried by $Y$ from [step 2.1]. Extend $\nu$ by zero off $K$, restrict both measures to $Y$, and extend $m$ by $1$ on $Y\setminus K$, which is null. Identify the integrals over $K$ and $Y$ by restriction and zero extension, since $\nu (K\setminus Y)=0$. Hence $\nu$ and ${\nu }_0$ are mutually absolutely continuous finite Borel measures on the standard Borel space $Y$ and have Radon–Nikodym densities; the isometry $J:L^2(Y,{\nu }_0;{\mathbb{C}}^m)\to L^2(Y,\nu ;{\mathbb{C}}^m)$, $J\xi =\sqrt{d{\nu }_0\mathrm{/}d\nu }\xi$, is unitary and commutes with every bounded Borel scalar multiplier by [F6]. Thus $U^{\mathrm{\prime }}:=J^{-1}\circ U$ is a unitary $H\to {\int }_Y^{\oplus }{\mathbb{C}}^{m(t)}d{\nu }_0(t)$ with $U^{\mathrm{\prime }}\mathcal{A}{U}^{\mathrm{\prime }-1}=\{M_f\}$ and $U^{\mathrm{\prime }}S{U}^{\mathrm{\prime }-1}=M_t$.

Transport along $c$: the map $c:X\to Y$ is a bimeasurable bijection with $c_{\ast }\mu ={\nu }_0$, so by [F7] pullback of sections is a unitary $T:{\int }_Y^{\oplus }{\mathbb{C}}^{m(t)}d{\nu }_0(t)\to {\int }_X^{\oplus }{\mathbb{C}}^{m(c(x))}d\mu (x)$ intertwining multiplication by $f$ with multiplication by $f\circ c$. Define $m^{\mathrm{\prime }}(x):=m(c(x))$, a Borel function $X\to \{1,2,\dots \}\cup \{\mathrm{\infty }\}$ by [F8].

The composite $W:=T\circ U^{\mathrm{\prime }}$ is a unitary $H\to {\int }_X^{\oplus }{\mathbb{C}}^{m^{\mathrm{\prime }}(x)}d\mu (x)$, and for every Borel $B\subseteq X$, $WP(B)W^{-1}=T{U}^{\mathrm{\prime }}{E}_S(c(B))U^{\mathrm{\prime }-1}{T}^{-1}=T{M}_{{\mathbf{1}}_{c(B)}}{T}^{-1}=M_{{\mathbf{1}}_{c(B)}\circ c}=M_{{\mathbf{1}}_B}$, using $P(B)=E_S(c(B))$ from [step 2.1] and the intertwining property of $T$.

Steps 1.1, 5.1 and 6.1 produce the faithful finite measure $\mu$, the Borel multiplicity $m^{\mathrm{\prime }}$ and the unitary $W$ with $WP(E)W^{-1}=M_{{\mathbf{1}}_E}$; any two $P$-faithful measures are mutually absolutely continuous by [step 1.1]. The uniqueness of the multiplicity is the rigidity statement of the intertwiner lemma: two models over the same base with a unitary intertwining all multiplications have the same multiplicity almost everywhere, and such an intertwiner is decomposable with unitary fibres a.e. (Unitary intertwiners preserve fibre multiplicity over a standard Borel base).