Haar null classes and Borel descent on a homogeneous space

Extension of the Weil formula to Borel sets: for every Borel $E\subseteq G$, ${\int }_{G\mathrm{/}H}{\int }_H{\mathbf{1}}_E(xh)dhd{\mu }_{\rho }(x)={\int }_E\rho (t)dt\mathrm{.}$ Both sides are $\sigma$-finite Borel measures on the $\sigma$-compact space $G$ that are finite on compacta (the right side because $\rho$ is continuous; the left side by sandwiching indicators of a compact set between $C_c$ functions). They agree on $C_c(G)$ by [F2], so by [F3] they agree on every Borel set.

Descent, first reduction: let $F$ be as in the statement. The set $N=\{(x,h,k):F(x,hk)\mathrm{\ne }F(x,h)\}$ is a Borel subset of the triple product, and its measure is zero: by Tonelli its measure is the integral over $k$ of the measures of the sections $N_k=\{(x,h):F(x,hk)\mathrm{\ne }F(x,h)\}$, each of which is null by hypothesis. Hence Fubini gives that for a.e. $x$ the section $N_x$ is a null subset of $H\times H$.

The coordinate map pushes the product measure to $\rho dt$: by [step 1.1] and left invariance of $dh$ (which lets $s(x)$ be replaced by any representative of the coset in ${\int }_H{\mathbf{1}}_E(\cdot h)dh$), for every Borel $E\subseteq G$ one has $({\mu }_{\rho }\otimes dh)({\mathrm{\Theta }}^{-1}(E))={\int }_{G\mathrm{/}H}{\int }_H{\mathbf{1}}_E(s(x)h)dhd{\mu }_{\rho }(x)={\int }_E\rho (t)dt$. Since $0<\rho <\mathrm{\infty }$ pointwise, the classes of $\rho dt$ and $dt$ coincide; hence the product class maps to the Haar class and, by [F1], $q^{-1}(E)$ is Haar null iff $({\mu }_{\rho }\otimes dh)(E\times H)=0$ iff ${\mu }_{\rho }(E)=0$.

For such an $x$, the homeomorphism $(h,k)\mapsto (h,hk)$ preserves the completed product $dh\otimes dh$ by [F4], so its image of $N_x$, namely $\{(h_1,h_2):F(x,h_2)\mathrm{\ne }F(x,h_1)\}$, is null. Hence $F_x$ is $dh$-a.e. constant for a.e. $x$.

Equivalence of two rho measures and of any quasi-invariant measure with ${\mu }_{\rho }$: if $\mu$ is a nonzero $\sigma$-finite quasi-invariant Borel measure on $G\mathrm{/}H$, replace it by an equivalent probability (still written $\mu$), choose a probability $\nu =w(t)dt$ with $w>0$ Haar-a.e., and set $\bar{\mu }(E)={\int }_G\mu (g^{-1}E)d\nu (g)$ for Borel $E$. The integrand is Borel in $g$ for fixed $E$, and $\bar{\mu }$ is a probability on $G\mathrm{/}H$. Each translate $g_{\ast }\mu (E)=\mu (g^{-1}E)$ is equivalent to $\mu$, so $\bar{\mu }\sim \mu$; and by Tonelli $\bar{\mu }(E)={\int }_{G\mathrm{/}H}\nu \{g:gx\in E\}d\mu (x)$. For fixed $x$ with representative $t=s(x)$, the substitution $g\mapsto gt$ gives $\nu \{g:gx\in E\}={\mathrm{\Delta }}_G(t{)}^{-1}{\int }_{q^{-1}(E)}w(u{t}^{-1})du$ by the right-translation scaling of the Haar integral; since $w>0$ Haar-a.e. and $q^{-1}(E)$ is right-$H$-invariant, this is positive exactly when ${\mu }_{\rho }(E)>0$ by [step 2.1]. Hence $\bar{\mu }(E)=0$ iff ${\mu }_{\rho }(E)=0$, so $\mu \sim {\mu }_{\rho }$. This proves the first assertion and, with [step 2.1], the null-class assertion $\mu (E)=0\iff q^{-1}(E)$ Haar null.

Borel selection of the constant: fix a strictly positive integrable Borel probability density $q$ on $H$: take a countable compact cover $(K_n)$, each of finite Haar measure, and normalize ${\sum }_n{2}^{-n}(1+\mathrm{\mid }{K}_n\mathrm{\mid }{)}^{-1}{\mathbf{1}}_{K_n}$, which is positive everywhere and has finite nonzero integral and a finite or countable orthonormal basis $(e_i)$ of $K$ by [F5]; set $b_{ij}(x)={\int }_H⟨F(x,h)e_i,e_j⟩q(h)dh$. Each $b_{ij}$ is Borel in $x$ by Tonelli, and for a.e. $x$ the matrix $(b_{ij}(x))$ is the matrix of the a.e. constant unitary value of $F_x$, hence unitary. The set of $x$ where the countably many Borel unitary equations ${\sum }_j{b}_{ij}\stackrel{‾}{b_{i^{\mathrm{\prime }}j}}={\delta }_{i{i}^{\mathrm{\prime }}}$ and column completeness fail is Borel and null; put $B(x)=I$ there and $B(x)$ equal to the operator with matrix $(b_{ij}(x))$ otherwise. Then $B:G\mathrm{/}H\to U(K)$ is Borel and $F(x,h)=B(x)$ for a.e. $(x,h)$.

Collecting the steps: every nonzero $\sigma$-finite quasi-invariant $\mu$ is equivalent to ${\mu }_{\rho }$ [step 3.1]; the coordinate map carries the product class to the Haar class, so a Borel $E\subseteq G\mathrm{/}H$ is $\mu$-null exactly when its full preimage $q^{-1}(E)$ is Haar null [step 2.1, step 3.1]; and every $H$-invariant Borel $U(K)$-valued function descends to a Borel $B$ almost everywhere [step 3.2]. These are the three assertions of the statement.