Exercise 18, § 1

Let X be a locally compact space, G a compact group operating continuously on X, E the orbit of a point of X, and $\mathscr{F}$ a vector space of continuous numerical functions on X such that, for every function $f \in \mathscr{F}$ and every $s \in G$, one has $\gamma(s)f \in \mathscr{F}$; assume in addition that $\mathscr{F}$ contains the constant functions on X. Let $x_0 \in X$ be a point invariant under G and such that $|f(x_0)| \leq \sup_{y \in E} |f(y)|$ for every function $f \in \mathscr{F}$. Show that one then has $f(x_0) = \int_G f(s \cdot z)\, ds$ for every $z \in E$ and every $f \in \mathscr{F}$. (Show that there exists a positive measure $\nu$ on E, of total mass 1, such that $f(x_0) = \int_E f(z)\, d\nu(z)$ for every $f \in \mathscr{F}$, and apply the Lebesgue–Fubini theorem.) The case that $X = \mathbf{R}^n$, G is the orthogonal group, and $x_0 = 0$; apply the formula (7) of §2.