
\begin{proof}~
\begin{align*}
  \operatorname{Vol}(\OO^{\times}, dx^{\times}) &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} \frac{dx}{|x|} \\
  &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} dx \\
  &= \frac{1}{1 - q^{-1}} \operatorname{Vol}(\OO^{\times}, dx) \\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  \operatorname{Vol}(\varpi \OO, dx) \big)\\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  |\varpi|\operatorname{Vol}(\OO, dx) \big)\\
  &= \operatorname{Vol}(\OO, dx)
\end{align*}
\end{proof}

\begin{defi}
  Soit $C_c^{\infty}(F^*)$ l'espace des applications $F^* \longrightarrow \mathbb{C}$ localement constantes à support compact.
\end{defi}


Preuve : 
\begin{align*}
\operatorname{Vol}\left(O^x, dx^x\right) &= \frac{1}{1-q^{-1}} \int_{O^x} \frac{dx}{|x|} \\
&= \frac{1}{1-q^{-1}} \int_{O^x} dx \\
&= q^{-1} \operatorname{Vol}\left(O_0, dx\right) \\
&= \mid \operatorname{Vol}(O, dx) \\
&= \frac{1}{1-q^{-1}}\left(\operatorname{Vol}(O, dx)-\overline{\operatorname{Vol}(O, dx)}\right) \\
&= \operatorname{Vol}(O, dx)
\end{align*}

- DEF : $C_c^{\infty}\left(F^x\right)$ est l'espace des applications $F^x \longrightarrow \mathbb{C}$ localement constantes sur $F^x$ à support compact.
