Exercise 31, § 4

Let $n$ be an integer $\geqslant 1$. Let N denote the euclidean norm on $\mathbf{R}^n$, and let $\lambda_n$ denote Lebesgue measure on $\mathbf{R}^n$. Let $\omega_n$ denote the unique positive measure of total mass 1 on the sphere $\mathbf{S}_n\subset \mathbf{R}^{n+1}$ invariant under the action of the orthogonal group $\mathbf{O}(n+ 1,\mathbf{R}) ($cf. INT, VIII, p. 116, exercise 8).

We identify $\mathbf{R}^n$ with the complement of a projective hyperplane H of the projective space $\mathbf{P}_n(\mathbf{R}) ($cf. TG, VI, p. 15, prop. 5), and measures on $\mathbf{R}^n$ will be interpreted as measures on $\mathbf{P}_n(\mathbf{R})$ carried by the complement of H.

For every $g\in \mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$, let $\widetilde{g}$ denote the homeomorphism of $\mathbf{P}_n(\mathbf{R})$ defined by $g$.

a) The positive measure $c_n$ on $\mathbf{R}^n$ defined by

("Cauchy measure") is bounded and of total mass 1. If $n= 1$, it is the stable measure with parameter 1 (cf. exercise V, p. 496). For every projective hyperplane $H'$ of $\mathbf{P}_n(\mathbf{R})$, one has $c_n(H') = 0$.

b) The canonical mapping of $\mathbf{R}^{n+1}-\{0\}$ into $\mathbf{P}_n(\mathbf{R})$ defines, by passing to subspaces, a continuous surjective mapping $\sigma :\mathbf{S}_n-(\mathbf{R}^n\times \{0\})\rightarrow \mathbf{R}^n$. One has $\sigma (\omega_n) =c_n$.

c) Let $g\in \mathbf{G}\mathbf{L}(n+1,\mathbf{R})$. There exist $a\in \mathbf{G}\mathbf{L}(n,\mathbf{R})$ and $b\in \mathbf{R}^n$ such that the image measure $\widetilde{g}(c_n)$ is equal to the image of $c_n$ under the affine mapping $x\mapsto a(x) +b$.

In what follows, the aim is to prove a converse of this assertion. d) Let $\mu$ be a bounded positive measure on $\mathbf{P}_n(\mathbf{R})$ such that every projective hyperplane is $\mu$-negligible. The set of $g\in \mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$ such that $g(\mu) =\mu$ and $|$det($g$)$|= 1$ is a compact subgroup of $\mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$. e) Let $\mu$ be a bounded positive measure on $\mathbf{P}_n(\mathbf{R})$ of total mass 1 such that, for every $g\in \mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$, there exists an affine mapping $f$ of $\mathbf{R}^n$ such that $g(\mu) =f(\mu)$. Every projective hyperplane is $\mu$-negligible.

f) Let $K_\mu$ be the subgroup of $g\in \mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$ such that $g(\mu) =\mu$. There exists $h\in \mathbf{G}\mathbf{L}(n+ 1,\mathbf{R})$ such that $K_\mu\subset h\mathbf{O}(n+ 1,\mathbf{R})h^{-1}$.

g) Let $K =h^{-1}K_\mu h$. One has $\mathbf{O}(n+ 1,\mathbf{R}) = G\cdot K$, where G is the subgroup of $\mathbf{O}(n+ 1,\mathbf{R})$ consisting of the matrices

0 1

with $x\in \mathbf{O}(n,\mathbf{R})$.

h) The measure $h^{-1}(\mu)$ coincides with $c_n$. (Observe that the group K acts transitively on $\mathbf{S}_n$ and that the image measure $\sigma (h^{-1}(\mu))$ is a K-invariant measure on $\mathbf{S}_n.$)