Spectral multiplicity model of a transitive system of imprimitivity

Choose a $P$-faithful finite Borel measure ${\mu }_0$ on $X$ by [F2] and apply the multiplicity model: there are a Borel $m:X\to \{1,2,\dots \}\cup \{\mathrm{\infty }\}$ and a unitary $W:H_0\to {\int }_X^{\oplus }{\mathbb{C}}^{m(x)}d{\mu }_0(x)$ with $WP(E)W^{-1}=M_{{\mathbf{1}}_E}$ for every Borel $E$. By [F5] ${\mu }_0$ is quasi-invariant, so ${\mu }_0$ lies in the normalized class of $G\mathrm{/}H$.

Conjugate the representation: $T_g:=W{U}_g{W}^{-1}$ is a unitary of the model with $T_g{M}_f{T}_g^{-1}=M_{f\circ g^{-1}}$ for every bounded Borel $f$, because $W$ conjugates $P(E)$ to $M_{{\mathbf{1}}_E}$ and $U_{g}P(E)U_g^{-1}=P(gE)$.

For each $g$, form the unitary ${\mathrm{\Theta }}_g:=c_g^{\ast }{T}_g$ where $c_g^{\ast }$ is the transport unitary associated with the base homeomorphism $x\mapsto gx$; here $c_g^{\ast }$ sends $\eta$ to $x\mapsto \eta (gx)$, from the model over $(X,{\mu }_0;m)$ to the pulled-back model over $(X,(g^{-1}{)}_{\ast }{\mu }_0;m\circ g)$, and intertwines $M_{f\circ g^{-1}}$ with $M_f$, so ${\mathrm{\Theta }}_g$ is a unitary $L^2(X,{\mu }_0;m)\to L^2(X,(g^{-1}{)}_{\ast }{\mu }_0;m\circ g)$ with ${\mathrm{\Theta }}_g{M}_f=M_f{\mathrm{\Theta }}_g$. Since $(g^{-1}{)}_{\ast }{\mu }_0$ is equivalent to ${\mu }_0$ by [F5], the Radon–Nikodym isometry $J_g$ of [F4] converts it into a unitary $U_g^{\mathrm{\prime }}:=J_g{\mathrm{\Theta }}_g:L^2(X,{\mu }_0;m)\to L^2(X,{\mu }_0;m\circ g)$ with $U_g^{\mathrm{\prime }}{M}_f=M_f{U}_g^{\mathrm{\prime }}$ for all bounded Borel $f$.

By the rigidity lemma [F3] applied to $U_g^{\mathrm{\prime }}$, the multiplicities agree: $m=m\circ g$ ${\mu }_0$-almost everywhere, for every $g$ (replacing $g$ by $g^{-1}$ gives the form stated in the strategy). Therefore each level set $\{m=k\}$ is invariant under the action up to ${\mu }_0$-null sets.

Ergodicity forces one level set to be conull: the countably many level sets partition $X$, each is invariant up to null sets, so by [F1] each is null or conull; since ${\mu }_0$ is nonzero and finite, exactly one level set $X_0=\{m=k_0\}$ is conull, and $k_0\in \{1,2,\dots \}\cup \{\mathrm{\infty }\}$. Restrict the model to $X_0$: the restriction of $W$ is a unitary $H_0\to L^2(X_0,{\mu }_0;k_0)$ and, viewed on $X$ by zero extension outside $X_0$, a unitary $W_0:H_0\to L^2(X,{\mu }_0;K)$ with $K={\mathbb{C}}^{k_0}$ ($K={\mathrm{\ell }}^2$ if $k_0=\mathrm{\infty }$), nonzero and separable, and $W_{0}P(E)W_0^{-1}=M_{{\mathbf{1}}_E}$ for every Borel $E$.

This proves existence with constant multiplicity and quasi-invariant $\mu ={\mu }_0$. Uniqueness: if $(W_1,{\mu }_1,K_1)$ and $(W_2,{\mu }_2,K_2)$ are two such normalizations, $W_2{W}_1^{-1}$ is a unitary intertwining the two multiplication actions over any common base; taking ${\mu }_1$ as the base and using mutual absolute continuity, [F3] gives $K_1\cong K_2$ isometrically and identifies the intertwiners as decomposable with unitary fibres, while ${\mu }_1\sim {\mu }_2$ by mutual absolute continuity of $P$-faithful measures.

Steps 4.1 and 5.1 establish the model and the constancy of the multiplicity, and step 6.1 gives the stated uniqueness; the measure is quasi-invariant by [step 1.1].