Proposition 9

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

Integration, VIII, p. 39, § 4. Convolution of measures and functions

Let $f, g$ be in $\mathscr{L}(G)$ . Assume that the function $s \mapsto g(s^{-1}x)f(s)\chi(s^{-1})$ is essentially $\beta$ -integrable except for a locally $\beta$ -negligible set of values of $x$ , and that the function

x \mapsto \int |g(s^{-1}x)f(s)|\chi(s^{-1})\, d\beta(s),

defined locally $\beta$ -almost everywhere, is locally $\beta$ -integrable. Then $f$ and $g$ are convolvable.

This follows from Prop. 1 of No. 1.

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

Cited by

Proposition 15, § 4

Cites

Prop. 1