Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups

Fix a compatible complete metric $d$ on $G$ by [F2]. By second countability and [F3], choose a countable base $(V_k{)}_{k\in \mathbb{N}}$ of relatively compact open sets such that for every open $U\subseteq G$, every $g\in U$, and every $ϵ>0$ there is a $V_k$ with $g\in V_k\subseteq \overline{V_k}\subseteq U$ and $\mathrm{diam}{V}_k<ϵ$: for each basic open set and each positive integer $m$, the refinements supplied by [F3] cover that basic open set, and second countability gives a countable subcover. By [F6] choose a point $c_k\in V_k$ for every $k$.

For $x\in G\mathrm{/}H$ define indices $k_n(x)$ by recursion: $k_0(x)$ is the least $k$ with $x\in q(V_k)$ and $\mathrm{diam}{V}_k\le 1$; such an index exists by choosing any $g\in q^{-1}(x)$ and applying [F3] inside $G$. Having defined $k_n(x)$, let $k_{n+1}(x)$ be the least $k$ with $\overline{V_k}\subseteq V_{k_n(x)}$, $x\in q(V_k)$, and $\mathrm{diam}{V}_k\le 2^{-(n+1)}$. This index exists by choosing $g\in q^{-1}(x)\cap V_{k_n(x)}$ and applying [F3] inside $V_{k_n(x)}$ with the prescribed diameter. Each $k_n$ is Borel: for $n=0$ each candidate condition on $x$ is membership in the open set $q(V_k)$, and inductively the set of $x$ with $\overline{V_k}\subseteq V_{k_n(x)}$ is the union over countably many $j$ with $\overline{V_k}\subseteq V_j$ of $\{x:k_n(x)=j\}$; intersecting with $q(V_k)$ and the fixed diameter condition gives Borel candidate sets. Selecting the least index in a Borel family is Borel.

The points $p_n(x):=c_{k_n(x)}$ are Borel functions of $x$ and converge: for $m\ge n$, $p_m(x)\in V_{k_m(x)}\subseteq \overline{V_{k_n(x)}}$, so $d(p_m(x),p_n(x))\le \mathrm{diam}\overline{V_{k_n(x)}}\le 2^{-n}$. Completeness gives a limit $s(x)\in {\bigcap }_n\overline{V_{k_n(x)}}$, whose diameter is zero. The map $s$ is Borel: for each nonempty closed $F\subseteq G$, $s(x)\in F$ iff $d(p_n(x),F)\to 0$; this condition is a countable intersection of countable unions of Borel sets because $p_n$ is Borel and distance to $F$ is continuous. The empty closed set has empty preimage.

$s$ is a cross-section: for each $n$ choose $g_n\in V_{k_n(x)}$ with $q(g_n)=x$, which is possible because $x\in q(V_{k_n(x)})$ by [step 1.2]. Both $g_n$ and $s(x)$ lie in $\overline{V_{k_n(x)}}$, a set of diameter at most $2^{-n}$, so $d(g_n,s(x))\le 2^{-n}\to 0$ and the sequence $g_n$ converges to $s(x)$; continuity of $q$ gives $x=q(g_n)\to q(s(x))$, and since the left-hand sequence is constant, $q(s(x))=x$.

Transport of measures: for a $\sigma$-finite Borel measure $\mu$ on $G\mathrm{/}H$, define $\nu (E):=\mu (s^{-1}(E))$ for Borel $E\subseteq G$. This is a Borel measure because $s$ is Borel. It is carried by the Borel image $s(G\mathrm{/}H)=\{g\in G:s(q(g))=g\}$: the latter is Borel since $g\mapsto (s(q(g)),g)$ is Borel and the diagonal of the Polish space $G$ is closed. If $G\mathrm{/}H={\bigcup }_n{A}_n$ with $\mu (A_n)<\mathrm{\infty }$, then $s(A_n)=s(G\mathrm{/}H)\cap q^{-1}(A_n)$ is Borel and $\nu (s(A_n))=\mu (A_n)$, since $q\circ s=\mathrm{i}\mathrm{d}$. The sets $s(A_n)$ together with $G\setminus s(G\mathrm{/}H)$ cover $G$, and the last set has $\nu$-measure zero, so $\nu$ is $\sigma$-finite. Equivalent measures have equivalent pushforwards by the definition of $\nu$, so the measure class is transported along $s$.

For $g\in G$ and $x\in G\mathrm{/}H$ one has $q(gs(x))=gq(s(x))=gx=q(s(gx))$ by [step 3.1], so $s(gx{)}^{-1}gs(x)\in q^{-1}(eH)=H$. Thus $h(g,x)=s(gx{)}^{-1}gs(x)$ is the unique element of $H$ with $gs(x)=s(gx)h(g,x)$. The map $(g,x)\mapsto h(g,x)$ is Borel as a composite of the Borel map $(g,x)\mapsto s(gx)$ (composition of $s$ with the continuous action map) with the continuous group operations.

The map $\mathrm{\Phi }:G\to G\mathrm{/}H\times H$, $g\mapsto (q(g),s(q(g){)}^{-1}g)$, is Borel and so is $\mathrm{\Psi }:G\mathrm{/}H\times H\to G$, $(x,h)\mapsto s(x)h$; they are mutually inverse: $q(s(x)h)=q(s(x))=x$ and $s(x{)}^{-1}s(x)h=h$, while $s(q(g))(s(q(g){)}^{-1}g)=g$. Hence $\mathrm{\Phi }$ is a Borel isomorphism onto $G\mathrm{/}H\times H$.

We have constructed a Borel cross-section $s$ of $q$ [step 3.1], the unique Borel section cocycle $h(g,x)=s(gx{)}^{-1}gs(x)$ [step 4.2], the Borel isomorphism $G\cong G\mathrm{/}H\times H$ [step 4.3], and the transported $\sigma$-finite measure [step 4.1]. AC is inherited through the Polish-space input [F1], whose complete-metrization proof uses DC and the ultrafilter lemma, and supplies Countable Choice for the countable-base refinements and base points $c_k$ in step 1.1 and the fibre points $g_n$ in step 3.1. The least-index recursion and the Borel cocycle and product formulas add no choice requirement.