Unitary equivalence of systems of imprimitivity and of the induced representations

Two systems of imprimitivity $(U,P)$ on $H$ and $(U^{\mathrm{\prime }},P^{\mathrm{\prime }})$ on $H^{\mathrm{\prime }}$ for the same Borel $G$-space $X$ (Systems of imprimitivity for a Borel $G$-space) are unitarily equivalent when there is a unitary $W:H\to H^{\mathrm{\prime }}$ (Hilbert space) with $W{U}_g{W}^{-1}=U_g^{\mathrm{\prime }},WP(E)W^{-1}=P^{\mathrm{\prime }}(E)$ for every $g\in G$ and every Borel $E\subseteq X$; if both are transitive on $X=G\mathrm{/}H$ (Transitive systems of imprimitivity and their normalized measure class), such a $W$ is called an equivalence of transitive systems. Two strongly continuous unitary representations $\sigma ,{\sigma }^{\mathrm{\prime }}$ of a common group (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) are unitarily equivalent when there is a unitary intertwiner between them.

Well-definedness. For a fixed base $X=G\mathrm{/}H$ the relation is the natural isomorphism of pairs (strongly continuous unitary representation, projection-valued measure): it is reflexive with $W=I$, symmetric with $W^{-1}$, and transitive with a composite, because conjugation by a unitary preserves the defining identities; the base isomorphism is suppressed from the notation exactly because transitivity fixes the identification $X=G\mathrm{/}H$. The definition introduces no choice and no new existence assertion.