Transitive systems of imprimitivity and their normalized measure class

A system of imprimitivity $(U,P)$ on a standard Borel $G$-space $X$ (Systems of imprimitivity for a Borel $G$-space, Standard Borel spaces) is transitive when $X$ is $G$-equivariantly isomorphic to a homogeneous space $G\mathrm{/}H$ with $H\le G$ closed, the isomorphism carrying the Borel structure of $G\mathrm{/}H$ (Left group actions, transitive actions, and faithful actions, Left and right cosets $gH$ and $Hg$ of a subgroup, Topological group: multiplication and inversion are continuous); hence then $G$ is second-countable locally compact, the action on $G\mathrm{/}H$ is the left-coset action, and the stabilizer of the identity coset is $H$. For a transitive system one fixes the base identification $X=G\mathrm{/}H$.

Assume AC for the following normalized-measure existence and uniqueness assertions: a normalized representative is a strongly quasi-invariant Radon measure ${\mu }_{\rho }$ on $G\mathrm{/}H$ built from a rho-function $\rho$ (Rho-function for a closed subgroup, Quasi-invariant Radon measure on G/H, Existence of rho-functions and quotient measure classes). The normalized homogeneous measure class is the unique class of nonzero quasi-invariant Radon measures on $G\mathrm{/}H$; the system is called transitive on $G\mathrm{/}H$.

Well-definedness. The equivariant-isomorphism clause is a condition on the given system and selects the conjugacy class of $H$: if $x_0\in X$ is the image of the identity coset under a $G$-equivariant Borel isomorphism, then ${\mathrm{Stab}}_G(x_0)=H$ by the computation $gH=H⟺g\in H$ (Left and right cosets $gH$ and $Hg$ of a subgroup); conversely a homogeneous space $G\mathrm{/}H$ for second-countable locally compact $G$ and closed $H$ is a standard Borel $G$-space with Borel action (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Compact lifts and averaging onto C_c(G/H)). The normalized class exists and is unique: the rho-function theorem supplies a full-support strongly quasi-invariant Radon representative ${\mu }_{\rho }$ (Existence of rho-functions and quotient measure classes), two rho-functions give representatives in the same class (their densities ${\rho }_1\mathrm{/}{\rho }_2$ are positive continuous), and every nonzero $\sigma$-finite quasi-invariant Borel measure is equivalent to ${\mu }_{\rho }$ (Haar null classes and Borel descent on a homogeneous space); in particular the class does not depend on the chosen rho-function, on the normalization of Haar measure, or on the choice of the base-point identification. Consumers that use only the definitional term transitive do not consume the normalized-measure existence assertion.

The definition names no choice; AC is used exactly by the quoted rho-function and Haar-lift suppliers for existence and uniqueness of the normalized class.