Exercise 11, § 3

Let $G$ be the compact group $O(n, R)$, $\lambda$ the normalized Haar measure on $G$. For $0 < k < n$, let $H$ be the closed subgroup of $G$ leaving invariant (globally) the subspace $\mathbf{R}^k$ of $\mathbf{R}^n$; the homogeneous space $G/H$, equipped with its quotient topology, may be identified canonically with the Grassmannian $E = G_{n-1, k-1}(\mathbf{R})$ (GT, VI, §3, No. 6) of the $k$-dimensional linear subspaces of $\mathbf{R}^n$. There is a measure $\mu$ on $E$ invariant under $G$, determined up to a constant. For every subspace $P \in E$, let $\sigma_P$ be the image of the Lebesgue measure of $\mathbf{R}^k$ under an $s \in G$ such that $s \cdot \mathbf{R}^k = P$ (independent of the element $s \in G$ satisfying this relation). Show that $\mu$ can be chosen in such a way that the following property is satisfied: for every continuous function $f$ on $\mathbf{R}^n$ with compact support, and for every $P \in E$, one sets $F(P) = \int_P f(x) d\sigma(x)$; then

(If $P_0 = \mathbf{R}^k$, note that one can write $\int_E F(P) d\mu(P) = c \int_G F(s \cdot P_0) d\lambda(s)$ for a suitable constant $c$, and on the other hand,

with the notations of Exercise 8; finally, make use of Exer. 8 b).)