The regular translation system on $L^2({\mathbb{R}}^n)$: position, momentum and trivial stabilizer

For $n=0$, the base and group are singletons, $L^2({\mathbb{R}}^0)=\mathbb{C}$ with unit mass, $U=I$ and $P$ is the one-point PVM; induction from the trivial group gives this system, and there are no coordinate operators. Thus all claims hold in that case. Assume $n\ge 1$ for the remaining steps. Then $U$ is a strongly continuous unitary representation and $P$ is a projection-valued measure by [F1] and [F2]. Covariance is a direct computation from $(U_{t}f)(x)=f(x-t)$: $U_{t}P(E)U_t^{-1}=M_{{\mathbf{1}}_E(\cdot -t)}=P(t+E)$. The action is transitive and the stabilizer of the origin is $\{0\}$ by [F3], so the base is ${\mathbb{R}}^n\mathrm{/}\{0\}={\mathbb{R}}^n$ with the trivial subgroup.

For each $j$, $M_{x_j}$ has domain $\{f\in L^2:x_{j}f\in L^2\}$ and is an unbounded self-adjoint real multiplication operator by the multiplier result of [F5]. Its spectral projections are $M_{{\mathbf{1}}_{\{x:x_j\in A\}}}$ for Borel $A\subseteq \mathbb{R}$. They commute, and the joint multiplication PVM is $P$; the unbounded coordinate integral $\int x_{j}dP$ equals $M_{x_j}$ on its stated domain, while bounded Borel functions of the coordinates act by the bounded integrals of [F2].

$U$ is induced from the trivial representation of the trivial subgroup: by the $H=\{e\}$ clause of [F4], ${\mathrm{Ind}}_{\{0\}}^{{\mathbb{R}}^n}1$ is the left regular representation on $L^2({\mathbb{R}}^n)$, and the canonical system of that induction is the multiplication system $P$.

For Schwartz $f$, the difference quotient $(U_{t{e}_j}f-f)\mathrm{/}t$ tends in $L^2$ to $-{\mathrm{\partial }}_{j}f$, by the fundamental theorem of calculus and a Schwartz majorant. To identify the full domain, apply the unitary Fourier transform of [F5]: coordinate translation becomes multiplication by $e^{-2\pi it{\xi }_j}$ in the usual Fourier normalization. The derivative limit exists precisely when ${\xi }_j\widehat{f}\in L^2$: sufficiency follows from $\mathrm{\mid }(e^{-2\pi it{\xi }_j}-1)\mathrm{/}t\mathrm{\mid }\le 2\pi \mathrm{\mid }{\xi }_j\mathrm{\mid }$ and dominated convergence, and necessity from an almost-everywhere convergent subsequence of any $L^2$ limit of the quotients, whose pointwise limit is $-2\pi i{\xi }_j\widehat{f}$. Define $P_j={\mathcal{F}}_2^{-1}{M}_{2\pi {\xi }_j}{\mathcal{F}}_2$ on $D_j=\{f:{\xi }_j{\mathcal{F}}_{2}f\in L^2\}$. The multiplier theorem makes it self-adjoint and its transported exponential is $U_{t{e}_j}=e^{-it{P}_j}$. Thus $G_j=-i{P}_j$ and $T_j=-i{G}_j=-P_j$ on the full domain $D_j$. On Schwartz functions the Fourier differentiation identity gives $P_j=-i{\mathrm{\partial }}_j$, recovering $G_j=-{\mathrm{\partial }}_j$ and $T_j=i{\mathrm{\partial }}_j$ there. No pointwise derivative of a general $L^2$ class is used.

Standard Borel and Polish: by [F6] the metric $d_{\mathrm{\infty }}$ makes ${\mathbb{R}}^n$ complete with countable dense subset ${\mathbb{Q}}^n$, so ${\mathbb{R}}^n$ is Polish and its Borel space is standard Borel, while the singleton case $n=0$ uses the zero metric as in step 1.1; the base with its standard Borel structure is the one used by the system.

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 verify all the displayed claims: transitivity with trivial stabilizer, the PVM as joint spectral measure of position, the induced-representation identification, the momentum generators, and the standard-Borel base.