Exercise 4, § 2

Let $X$ be a locally compact space in which a locally compact group $H$ acts on the right continuously and properly. Let $\beta$ be a left Haar measure on $H$, $\mu$ a measure $\geqslant 0$ on $X$. Let $h$ be a continuous function $\geqslant 0$ on $X$ such that $h^b = 1$. If $f$ is a $\mu$-negligible function $\geqslant 0$ on $X$, the function $(x, \xi) \mapsto f(x)h(x\xi)$ is $(\mu \otimes \beta)$-negligible on $X \times H$. (There exist decreasing open sets $\Omega_1, \Omega_2, \ldots$ such that $\mu(\Omega_n)$ tends to 0 and such that $f$ is zero outside $\Omega_1 \cap \Omega_2 \cap \cdots$. Show that $\int^* \varphi_{\Omega_n}(x)h(x\xi) d\mu(x)d\beta(\xi) \leqslant \mu(\Omega_n)$, on observing that the function $(x, \xi) \mapsto \varphi_{\Omega_n}(x)h(x\xi)$ is lower semi-continuous.)