Exercise 35, § 9

Let $G$ be a compact subgroup of $\mathbf{GL}(n, \mathbf{R})$. (a) Let $A, B$ be disjoint compact $G$-invariant subsets of $\mathbf{R}^n$. There exists a $G$-invariant polynomial function $u$ on $\mathbf{R}^n$, such that $u = 1$ on $A$ and $u = 0$ on $B$. (There exists a continuous real-valued function $v$ on $\mathbf{R}^n$ such that $v = 1$ on $A$ and $v = 0$ on $B$. By the Stone-Weierstrass Theorem, there exists a polynomial function $w$ on $\mathbf{R}^n$ such that $|v - w| \leq \frac{1}{3}$ on $A \cup B$. Derive $u$ by an integration process with respect to the normalized Haar measure of $G$.) (b) Let $x \in \mathbf{R}^n$. Then $Gx$ is the set of zeros in $\mathbf{R}^n$ of a finite number of $G$-invariant polynomials. (Use (a) and Corollary 2 to Theorem 1, no. 8.)