Lemma 9

Let $\psi$ be a polynomial $\neq 0$ of $K[X_1, \ldots, X_r]$. In the space $K^r$, the set N defined by $\psi(x_1, \ldots, x_r) = 0$ is negligible for the Haar measure.

Let us argue by induction on $r$. The lemma is evident for $r = 1$, since N is then a finite set. Changing if necessary the numbering of the variables, we can suppose that $\psi \notin K[X_1, \ldots, X_{r-1}]$; write

with $m > 0$ and $\psi_0 \neq 0$. In the space $K^{r-1}$, let $N_0$ be the set defined by $\psi_0(x_1, \ldots, x_{r-1}) = 0$, which is negligible by the induction hypothesis. For every $(x_1, \ldots, x_{r-1}) \notin N_0$, the set of $x_r \in K$ such that $(x_1, \ldots, x_{r-1}, x_r) \in N$ is finite, therefore negligible. Since $K^r$ is countable at infinity (§ 1, No. 10, Cor. of Prop. 13), $N \cap [(K^{r-1} - N_0) \times K]$ is negligible in $K^r$ (Ch. V, § 8, No. 2, Prop. 4). Therefore N is negligible.