Corollary

Tag 09DX, permanent. Cite it as [Bourbaki, Tag 09DX].

Integration, chapter VII, § 1. Construction of a Haar measure

— Let f be a function defined on $K^*$ , with values in $\overline{\mathbf{R}}$ or in a Banach space. For f to be $\beta$ -*integrable, it is necessary and sufficient* that $(\mathrm{mod}_K)^{-1} f$ be $\alpha$ -integrable, in which case

\int_{K^*} f(x) d\beta(x) = \int_{K^+} (\mathrm{mod}_K(x))^{-1} f(x) d\alpha(x).

This follows from Prop. 14, the Cor. of Prop. 13, and Ch. V, §5, No. 3, Th. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

V, §5, No. 3, Th. 1
Cor. of Prop. 13
Prop. 14