Exercise 9, § 2

Let X be a locally compact space, H a compact group operating continuously (hence properly) on the right in X. Denote by $\beta$ the normalized Haar measure on H, by $\pi : X \to X/H$ the canonical mapping. Let $\lambda$ be a positive measure on X/H, $\lambda^\#$ the corresponding measure on X.

a) Show that if N is a $\lambda$-negligible subset of X/H, then $\overline{\pi}^{-1}(N)$ is $\lambda^\#$-negligible. (Calculate the measure of a set $\overline{\pi}^{-1}(U)$, where U is open in X/H.)

b) Let $p$ be a finite number $\geqslant 1$, and let $f$ be a function in $\mathscr{L}_F^p(X, \lambda^\#)$ (F a Banach space or $F = \overline{\mathbf{R}}$). Show that the set of $x \in X$ such that $\xi \mapsto f(x\xi)$ is not $\beta$-integrable on H is of the form $\overline{\pi}^{-1}(N)$, where N is $\lambda$-negligible. Moreover, if $f^b$ is the function on X/H, defined almost everywhere (for $\lambda$), such that $f^b(\pi(x)) = \int_H f(x\xi) d\beta(\xi)$, then $f^b$ belongs to $\mathscr{L}_F^p(X/H, \lambda)$, and $N_p(f^b) \leqslant N_p(f)$.

c) Conversely, if $g \in \mathscr{L}_F^p(X/H, \lambda)$, then the function $g \circ \pi$ belongs to $\mathscr{L}_F^p(X, \lambda^\#)$. If $p, q$ are conjugate exponents, $F'$ is the dual of F, $f \in \mathscr{L}_F^p(X, \lambda^\#)$ and $g \in \mathscr{L}_{F'}^q(X/H, \lambda)$, then