Essential uniqueness of the type I irreducible disintegration

In the zero-space case both measures are zero: otherwise the positive-measure support with nonzero irreducible amplified fibres would give a nonzero square-integrable localized fundamental section, contradicting the zero direct integral. The measures are then equivalent and the almost-everywhere multiplicity assertion is vacuous. For $H\mathrm{\ne }0$, every model satisfying the labelled decomposition conditions of [F1] is central on its conull standard-Borel support: repeat the intrinsic ideal-support and fibre-centre argument of [F1] for that model. The intrinsic ideal-range projections are scalar indicators of the same countable generating ideal-opens, hence the full diagonal algebra lies in the generated algebra; the fibre centre is scalar, so the centre is exactly that diagonal algebra. That argument uses finite-measure localizations and applies to sigma-finite measures as well as to the normalized probability measure used in the existence construction. Apply [F2] to obtain a conull bimeasurable map $c$ carrying the first measure to a measure equivalent to the second, and unitary equivalences between the factor fibre at $b$ in the first model and that at $c(b)$ in the second.

The two fibres are respectively $m(b)$ copies of the irreducible class $b$ and $n(c(b))$ copies of the irreducible class $c(b)$. Their unitary equivalence and [F3] force $b=c(b)$ and $m(b)=n(c(b))$ on one conull set. Thus the central base isomorphism is the identity on actual dual labels, so its pushforward measure-class assertion is precisely $\mu \sim \nu$ on the same dual. Their multiplicities coincide almost everywhere; transitivity makes the measure class and multiplicity equivalence class invariants of $\pi$. This use of labelled fibre equivalence is legitimate because centrality and class identification were proved in [F1]; it does not follow merely from using the same name for two bases.