Systems of imprimitivity for a Borel $G$-space

Let $G$ be a second-countable locally compact Hausdorff topological group (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Second countability: an at most countable basis for the topology, Topological group: multiplication and inversion are continuous) acting measurably on a standard Borel space $X$ (Standard Borel spaces), that is, the map $(g,x)\mapsto gx$ is $\mathcal{B}(G)\otimes \mathcal{B}(X)$-measurable (Left group actions, transitive actions, and faithful actions, A measurable function between measurable spaces, Measurable spaces and measurable sets), and let $H$ be a separable complex Hilbert space (Hilbert space, Separability: the existence of an at most countable dense subset). A system of imprimitivity for the action is a pair $(U,P)$ in which $U:G\to U(H)$ is a strongly continuous unitary representation (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) and $P$ is a projection-valued measure on the Borel $\sigma$-algebra of $X$ (Projection valued measure) such that

$U_{g}P(E)U_g^{-1}=P(gE)(g\in G,E\subseteq X\text{ Borel})\mathrm{.}$

The family of $P$-null Borel sets is the null-set class of the system; the system is ergodic when every Borel $E$ with $U_{g}P(E)U_g^{-1}=P(E)$ for all $g\in G$ satisfies $P(E)=0$ or $P(E)=I$, and is nonzero when $P(X)=I$ and $H\mathrm{\ne }\{0\}$.

Well-definedness. The covariance relation is a condition on the given pair: for fixed $g$ the map $E\mapsto P(gE)$ is a projection-valued measure because $E\mapsto gE$ is a $\sigma$-algebra automorphism of $\mathcal{B}(X)$, and $U_{g}P(E)U_g^{-1}$ is the projection $U_g(P(E)H)$ with the same range as $P(E)$ transported by the unitary $U_g$, so both sides of the displayed identity are orthogonal projections (Projection valued measure); since $U_g$ is unitary, $U_g^{-1}=U_g^{\ast }$ throughout. The null-set class is a $\sigma$-ideal of $\mathcal{B}(X)$: a projection $P(E)$ vanishes exactly when the finite scalar set functions $E\mapsto ⟨P(E)\xi ,\xi ⟩$ ($\xi \in H$) all vanish, and these are countably additive because the series in clause 4 of the projection-valued measure definition converges in norm and the inner product is continuous (Projection valued measure). No regularity of $P$ is assumed, no topological condition beyond measurability of the action is imposed, and the definition itself makes no choice.