Local densities for equivalent Radon quotient measures

Choose a relatively compact symmetric open identity neighborhood $U\subseteq G$ and set $L={\bigcup }_{n\ge 1}{U}^n$. Then $L$ is open and $\sigma$-compact. Its action on $X$ partitions $X$ into disjoint open orbits: the orbit through $xH$ is the image of $L$ under $\mathrm{\ell }\mapsto \mathrm{\ell }xH$, so it is $\sigma$-compact; it is LCH by [F2]. The compact closures of the sets $U^n$ give each orbit a countable compact cover.

Restrict $\mu$ and $\nu$ to one orbit. Each restriction is $\sigma$-finite because it is Radon and the orbit is a countable union of compact sets of finite measure. Equivalence gives ${\nu }_i\ll {\mu }_i$ and ${\mu }_i\ll {\nu }_i$; [F4] supplies a measurable $w_i$ with $d{\nu }_i=w_{i}d{\mu }_i$, where $0<w_i<\mathrm{\infty }$ almost everywhere. The RN uniqueness clause makes $w_i$ unique up to ${\mu }_i$-null sets.

By [A1] choose one measurable version separately on each orbit; all density operations below are performed componentwise. On each component, multiplication by $w_i^{-1\mathrm{/}2}$ maps $L^2({\mu }_i;V)$ isometrically onto $L^2({\nu }_i;V)$, since $\int \mathrm{\parallel }{w}_i^{-1\mathrm{/}2}F{\mathrm{\parallel }}^{2}d{\nu }_i=\int \mathrm{\parallel }F{\mathrm{\parallel }}^{2}d{\mu }_i$; its inverse is multiplication by $w_i^{1\mathrm{/}2}$. Taking the Hilbert direct sum of these componentwise unitaries gives the asserted map on the completed locally measurable section spaces.