Transport of central models and disintegration of intertwiners

Assume the Axiom of Choice. Let $G$ be a second-countable locally compact group and $(\pi ,H)$ a separable strongly continuous unitary representation with two central decompositions $U:H\to {\int }_X^{\oplus }{H}_{x}d\mu (x)$, $\pi \cong {\int }^{\oplus }{\pi }_{x}d\mu$, and $V:H\to {\int }_Y^{\oplus }{K}_{y}d\nu (y)$, $\pi \cong {\int }^{\oplus }{\sigma }_{y}d\nu$, in the sense of Central decomposition into factor representations. Then there exist conull Borel sets $X_0\subseteq X$, $Y_0\subseteq Y$, a bimeasurable bijection $c:X_0\to Y_0$ with $c_{\ast }(\mu {\mathrm{\mid }}_{X_0})$ equivalent to $\nu {\mathrm{\mid }}_{Y_0}$, and a field of unitaries $u_x:H_x\to K_{c(x)}$ that is measurable over $X_0$ and satisfies $u_x{\pi }_x(g)u_x^{-1}={\sigma }_{c(x)}(g)\text{for every }g\in G\text{ and }\mu \text{-almost every }x\in X_0\mathrm{.}$ Consequently the two central decompositions determine the same base modulo null sets and null-set modification, and the fibre representations are unitarily equivalent almost everywhere through a measurable field.

AC is inherited from central decomposition and spatialization and supplies the countable dense choice used in the common-null-set argument. Discarding zero fibre strata is permitted by the definition of central decomposition; null total spaces use empty conull bases. The group is second countable, and strong continuity is essential for extending from the countable dense set. The Radon–Nikodym weight affects norms and measure normalization but cancels from intertwining because it is scalar. No source citation replaces a local supplier proof. The referenced complete Bekka–de la Harpe PDF, pp. 195–202, and Blackadar PDF pp. 255–262 were consulted for the central/type-I architecture; Blackadar explicitly outlines the direct-integral theory and refers technical details elsewhere. The measurable and spatial steps here use the proved local suppliers named above.