Mackey Borel structure and countable separation of the unitary dual

On the one-dimensional carrier $H_1$, the constant map $g\mapsto I_{H_1}$ is a strongly continuous unitary representation: it is a homomorphism and every orbit map is constant. Its only closed linear subspaces are $\{0\}$ and $H_1$, so it is irreducible; consequently both $\widehat{G}$ and $\mathcal{I}(G)$ are nonempty.

Let $\pi$ be irreducible on a nonzero Hilbert space $H$ and fix $0\mathrm{\ne }\xi \in H$. The closed span $K=\stackrel{}{\mathrm{span}}\{\pi (g)\xi :g\in G\}$ is nonzero and invariant, because $\pi (h)$ maps the orbit bijectively to itself by $\pi (h)\pi (g)\xi =\pi (hg)\xi$ and is unitary; thus $K=H$ by [F3]. By [A1, F1, F2], choose an at-most-countable dense set $D\subseteq G$. Continuity of $g\mapsto \pi (g)\xi$ implies $\{\pi (d)\xi :d\in D\}$ is dense in the orbit: the inverse image of any neighborhood of $\pi (g)\xi$ is a neighborhood of $g$ and meets $D$. The complex span of this countable orbit is dense in $H$. By [F4], its finite $\mathbb{Q}+i\mathbb{Q}$-linear combinations form an at-most-countable set $V$. They are dense in the complex span: for any finite sum ${\sum }_{j=1}^m{c}_j\pi (d_j)\xi$ and $\epsilon >0$, choose $r_j\in \mathbb{Q}+i\mathbb{Q}$ with $\mathrm{\mid }{c}_j-r_j\mathrm{\mid }<\epsilon \mathrm{/}(m(1+\mathrm{\parallel }\pi (d_j)\xi \mathrm{\parallel }))$; the triangle inequality makes the resulting rational-complex sum differ by less than $\epsilon$. Thus $V$ is a countable dense subset of $H$, so $H$ is separable by [F5].

The set $V$ in step 1.2 is nonempty because it contains the empty sum $0$. By [F6], there is a sequence with range $V$; [F7], using the countable choice supplied by [A1, F1], gives a finite or countable orthonormal basis $E$ of $H$ and a unitary Fourier-coefficient map $\mathrm{\Phi }:H\to {\mathrm{\ell }}^2(E,\mathbb{C})$. Since $H\mathrm{\ne }0$, $E$ is nonempty, so its cardinal is some $n\in \{1,2,\dots ,{\mathrm{\aleph }}_0\}$. The coordinate vectors give ${\mathrm{\ell }}^2(E,\mathbb{C})$ a complete orthonormal basis by [F8, F9]; a complete orthonormal basis of $H_n$ has the same cardinality $n$ by [F7]. Reindex these two bases by bijections with $n$ and apply the Fourier-coefficient theorem [F7] to obtain a unitary $R:{\mathrm{\ell }}^2(E,\mathbb{C})\to H_n$. Then $U:=R\mathrm{\Phi }$ is unitary, and $\tilde{\pi }(g):=U\pi (g)U^{-1}$ is irreducible and strongly continuous: conjugation preserves invariant closed subspaces and preserves orbit-map norm continuity. Hence $\tilde{\pi }\in {\mathrm{Irr}}_n(G)$ and $q(\tilde{\pi })=[\pi ]$, so $q$ is onto.

The collection ${\mathcal{B}}_M(\widehat{G})=\{E\subseteq \widehat{G}:q^{-1}(E)\in \mathcal{B}(\mathcal{I}(G))\}$ is a sigma-algebra: inverse images preserve the whole set, complements, and countable unions. Thus it is the quotient Borel structure specified in the definition. A Borel space is countably separated exactly when a countable family of its Borel sets separates every pair of distinct points; applying this definition to $(\widehat{G},{\mathcal{B}}_M(\widehat{G}))$ gives the stated meaning, without asserting that it is countably separated or standard Borel.