Proposition 8

Assume $X/H$ paracompact. There exists a continuous function $h \geqslant 0$ on $X$, whose support has compact intersection with the saturation of every compact subset of X, and is such that $h^b = 1$. For such a function, one has $g = (h \cdot (g \circ \pi))^b$ for every continuous function $g$ on $X/H$.

Let us apply Prop. 1 of No. 1, with $\chi = 1$, on taking for $f$ a function $\geqslant 0$ that is not identically zero on any orbit. We have $f^1(x) > 0$ at every point $x$ of $X$. Set $h = f/f^1$. Then $h^1 = f^1/f^1 = 1$, therefore $h^b = 1$. It follows that if $g$ is a continuous function on $X/H$, then $(h \cdot (g \circ \pi))^b = h^b \cdot g = g$.