Mackey little-group reduction for an abelian normal subgroup

The representation space is separable even if this was not assumed. For $v\mathrm{\ne }0$, the closed span of $\pi (G)v$ is invariant and hence is the whole space by irreducibility. A countable dense subset $D\subseteq G$ exists because $G$ is second-countable LCH; strong continuity makes $\pi (D)v$ dense in the orbit. Its finite rational-complex linear combinations are countable and dense in the Hilbert space. Thus [F1] applies. It produces $P$, which is ergodic by [F2] and concentrates on a Borel orbit $C=K\cdot \chi$ by [F3].

The continuous orbit bijection $r:K\mathrm{/}{K}_{\chi }\to C$ of [F4] is bimeasurable. Indeed, every open subset $O$ of the second-countable LCH quotient is a countable union of compact sets contained in $O$: use a countable base with compact closures and shrink inside $O$. Their images under $r$ are compact, hence closed in the Hausdorff dual, so $r(O)$ is Borel. This proves measurability of $r^{-1}$ without assuming that $r$ is a homeomorphism. Transporting $P{\mathrm{\mid }}_C$ gives a transitive system for $G$ on $G\mathrm{/}{H}_{\chi }\cong K\mathrm{/}{K}_{\chi }$; $N$ acts trivially on the base. By [F5], $\pi \cong {\mathrm{Ind}}_{H_{\chi }}^G\sigma$ for a strongly continuous $\sigma$.

Choose the Borel section in $G$ with values $s(x)\in K$. For every $n\in N$, the spectral formula makes $\pi (n)$ multiplication by $x(n)$ on the orbit, and the reconstruction gauge commutes with this scalar multiplier. Since $N$ fixes the base, the induced Radon–Nikodym factor is one and the induced formula gives $\sigma (s(x{)}^{-1}ns(x))=\chi (s(x{)}^{-1}ns(x))I$ for a.e. $x$. Intersect these conull sets over a countable dense subset of $N$ and fix one $x$ in the intersection. Continuity of both sides extends the identity to all $n\in N$ at this $x$. Conjugation by $s(x)$ maps $N$ onto itself, so $\sigma (m)=\chi (m)I$ for all $m\in N$. Put $\theta =\sigma {\mathrm{\mid }}_{K_{\chi }}$; it is strongly continuous and $\sigma (n,k)=\chi (n)\theta (k)$. Stabilizer invariance of $\chi$ verifies multiplicativity of this formula in the semidirect product.

$\theta$ is irreducible: if $\theta$ had a nontrivial closed invariant subspace, inducing it would produce a nontrivial closed invariant subspace of ${\mathrm{Ind}}_{H_{\chi }}^G(\chi \otimes \theta )\cong \pi$, because the quotient measure class has full support so a nonzero fibrewise subspace induces a nonzero closed subspace; this contradicts irreducibility of $\pi$.

Therefore $\pi$ is unitarily equivalent to ${\mathrm{Ind}}_{H_{\chi }}^G(\chi \otimes \theta )$ with $\chi \in \widehat{N}$ and $\theta$ an irreducible strongly continuous unitary representation of $K_{\chi }$, as claimed; the orbit $\chi$ is the one selected by the spectral PVM, and the inducing class is determined by the system uniqueness theorem.