Bruhat cutoff normalized along H-fibers

Choose a relatively compact symmetric open identity neighborhood $U$ in $G$ and let $L={\bigcup }_{n\ge 1}{U}^n$. Then $L$ is an open subgroup and is $\sigma$-compact, since $L={\bigcup }_n(\bar{U}{)}^n$. Its orbits on $X$ are open and disjoint; each is a continuous image of $L$, hence $\sigma$-compact. As an open subspace of the LCH space $X$, each orbit is regular and Lindelöf. By [F1] and [F3], every orbit is paracompact, and their topological sum $X$ is paracompact.

Cover $X$ by relatively compact open sets. By [F4] choose a locally finite partition of unity $({\psi }_i)$ subordinate to this cover; each $\mathrm{supp}{\psi }_i$ is compact. Use [F5] to choose ${\chi }_i\in C_c(X)$ with ${\chi }_i=1$ on $\mathrm{supp}{\psi }_i$, and [F2] to choose a nonnegative $u_i\in C_c(G)$ with $T_H{u}_i={\chi }_i$. Define $b_i=({\psi }_i\circ p)u_i$. It is continuous, nonnegative and compactly supported, and $T_H{b}_i={\psi }_i{\chi }_i={\psi }_i$.

Set $\beta ={\sum }_i{b}_i$. Since $({\psi }_i)$ is locally finite and $p$ is continuous, the sum is locally finite on $G$, hence continuous and nonnegative. Fiber integration gives $T_H\beta ={\sum }_i{\psi }_i=1$. For compact $Q\subseteq X$, only finitely many $\mathrm{supp}{\psi }_i$ meet $Q$; the support of $\beta$ over $Q$ is contained in the finite union of the compact sets $\mathrm{supp}{u}_i\cap p^{-1}(Q)$. Thus it is compact. AC supplies the choices, and the construction applies to non-$\sigma$-compact $X$ because it uses the open $L$-orbits from step 1.1. ∎