The regular representation of the real line as a multiplicity-one integral of characters

By [F1], the assumed AC supplies both DC and countable choice. Thus the DC hypotheses in [F9] and the countable-choice hypotheses in [F6] are met; the direct-integral assumptions are also covered by the given AC.

The metric and compact intervals in [F2] make $\mathbb{R}$ Hausdorff and locally compact, and rational intervals give second countability. With the group operations verified in [F3], $\mathbb{R}$ is a second-countable LCA group. Therefore [F5] applies to show that $\widehat{\mathbb{R}}$ is LCH abelian.

Let $J$ initially act on Borel representatives by $Jf(\chi )=f({\chi }^{-1})$. Inversion preserves the dual Haar measure by [F10], and it is Borel by [F5], so composition preserves measurability and null equivalence. For every nonnegative Borel function $g$, the map $s\mapsto s\circ \iota$ bijects simple minorants of $g$ and $g\circ \iota$; their simple integrals agree because inversion preserves the measures of their level sets. Taking suprema in the definition of the nonnegative integral [F10] gives $\int g\circ \iota d\mu =\int gd\mu$. Applying this to $g=\mathrm{\mid }f{\mathrm{\mid }}^2$ proves that $J$ is a linear isometry on $L^2$. Since inversion is involutive, $J^2=I$, so $J$ is unitary. Put ${\mathcal{F}}_{+}:=J{\mathcal{F}}_{-}$; it is unitary by [F9]. For every $f$ and $t$, $J{M}_{\overline{\chi (t)}}f(\chi )=\overline{{\chi }^{-1}(t)}f({\chi }^{-1})=\chi (t)Jf(\chi )\mathrm{.}$

By Step 1.2, the LCA hypotheses in [F9] hold. Let $s$ be a simple function with finite-measure support. It lies in $L^1\cap L^2$; [F9] says the $L^2$ Plancherel transform ${\mathcal{F}}_{-}$ agrees there with the integral Fourier transform. Since $\lambda (t)s=T_{t}s$, the Fourier translation formula in [F9] gives ${\mathcal{F}}_{-}\lambda (t)s=M_{\overline{\chi (t)}}{\mathcal{F}}_{-}s$. Such simple functions are dense in $L^2$ by [F9], while $\lambda (t)$, $M_{\overline{\chi (t)}}$ and ${\mathcal{F}}_{-}$ are bounded by [F4, F9, F11], so the equality extends to every $f\in L^2(\mathbb{R})$.

By [F4], $\mathrm{\Phi }:\mathbb{R}\to \widehat{\mathbb{R}}$, $\mathrm{\Phi }(\xi )={\chi }_{\xi }$, is a bijection and a group homomorphism. The evaluation formula is jointly continuous: near $({\xi }_0,t_0)$, $\mathrm{\mid }\xi t-{\xi }_0{t}_0\mathrm{\mid }\le \mathrm{\mid }\xi \mathrm{\mid }\mathrm{\mid }t-t_0\mathrm{\mid }+\mathrm{\mid }{t}_0\mathrm{\mid }\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }$, and continuity of the complex exponential in [F4] then gives continuity of $e^{2\pi i\xi t}$. For a subbasic compact-open neighborhood $S(K,V)=\{\chi :\chi [K]\subseteq V\}$ containing ${\chi }_{{\xi }_0}$, this joint continuity and compactness of $K$ give finitely many product neighborhoods covering $\{{\xi }_0\}\times K$ on which the exponential remains in $V$; intersecting their parameter neighborhoods gives an interval around ${\xi }_0$ mapped into $S(K,V)$. The inverse is continuous at the identity character: for $\delta >0$, put $K_{\delta }=[-1\mathrm{/}(2\delta ),1\mathrm{/}(2\delta )]$ and $V_0=\{z\in \mathbb{T}:\mathrm{\mid }z-1\mathrm{\mid }<1\}$. If ${\chi }_{\xi }[K_{\delta }]\subseteq V_0$ and $\mathrm{\mid }\xi \mathrm{\mid }\ge \delta$, then $t=1\mathrm{/}(2\mathrm{\mid }\xi \mathrm{\mid })\in K_{\delta }$ and ${\chi }_{\xi }(t)=e^{\pm \pi i}=-1\mathrm{\notin }{V}_0$, a contradiction. Continuity of translations in the dual group from [F5] gives continuity of ${\mathrm{\Phi }}^{-1}$ everywhere. Hence $\mathrm{\Phi }$ is a homeomorphism.

The homeomorphism in Step 2.2 makes $\widehat{\mathbb{R}}$ second countable; it is LCH by Step 1.2. Thus [F5] gives its standard-Borel structure. The compact sets $K_n=\mathrm{\Phi }([-n,n])$ cover the dual, and [F5] gives $\mu (K_n)<\mathrm{\infty }$; hence $\mu$ is sigma-finite by [F5].

Set $e_0(\chi )=1$ and $e_n(\chi )=0$ for $n>0$. Its Gram coefficients are constant and its values span $\mathbb{C}$, so [F7] makes $(H_{\chi })$ the constant measurable Hilbert field. Every ${\pi }_{\chi }$ is a unitary homomorphism and is strongly continuous by [F4, F8]. For fixed $t$, the scalar field $\chi \mapsto \chi (t)$ is continuous by [F8], hence weakly measurable. The standard-Borel sigma-finite base was established in Step 3.1, so the in-run definition [F8] applies and forms $\mathrm{\Pi }={\int }^{\oplus }{\pi }_{\chi }d\mu (\chi )$.

The map $\mathcal{V}:L^2(\widehat{\mathbb{R}},\mu )\to {\int }_{\widehat{\mathbb{R}}}^{\oplus }\mathbb{C}d\mu$, $\mathcal{V}f=[\chi \mapsto f(\chi )]$, is a unitary by the quotient definition in [F7]. From the pointwise definition of the direct-integral representation in [F8], ${\mathcal{V}}^{-1}\mathrm{\Pi }(t)\mathcal{V}$ is multiplication by $\chi \mapsto \chi (t)$.

By Steps 1.3 and 2.1, ${\mathcal{F}}_{+}=J{\mathcal{F}}_{-}$ intertwines $\lambda (t)$ with multiplication by $\chi (t)$. By Step 5.1 this is $\mathrm{\Pi }(t)$ under the canonical direct-integral identification, so $\mathcal{V}{\mathcal{F}}_{+}$ intertwines the left regular representation with ${\int }^{\oplus }{\pi }_{\chi }d\mu$. Each fibre is exactly $\mathbb{C}$, and [F4] parametrizes each character exactly once. The vector ${\mathbf{1}}_{(0,1]}$ is nonzero in $L^2(\mathbb{R})$ because $m_{\mathbb{R}}((0,1])=1$ by [F6], so the decomposition is not the zero Hilbert space. The homeomorphic group isomorphism $\mathrm{\Phi }$ pulls $\mu$ back to a nonzero regular Borel measure finite on compact sets and invariant under translations, hence to a left Haar measure on $\mathbb{R}$ by [F6]. By [F12] it is a positive multiple of Lebesgue measure; its countable subsets are null, so the parameter measure has no point masses. This is the stated multiplicity-one continuous-spectrum model.