Proposition 14

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

Integration, chapter VIII, § 4. Convolution of measures and functions

— If $f \chi^{-1} \in L^1(G, \beta)$ and $g \in L^\infty(G, \beta)$ , then the formula (15) defines for every $x \in G$ a product $f * g$ that is bounded and is uniformly continuous for the right uniform structure of $G$ .

We already know that $f * g$ belongs to $L^\infty(G, \beta)$ (No. 2, Prop. 5); moreover, $(f * g)(x) = \int f(xs^{-1})g(s)\, d\nu(s)$ , on setting $\nu = {\chi'}^{-1} \cdot \beta$ ; $\nu$ is a right Haar measure. Therefore

|(f * g)(x) - (f * g)(x')| \leq \|g\|_\infty \int |f(xs^{-1}) - f(x's^{-1})|\, d\nu(s)

= \|g\|_\infty \int |(f(s^{-1}) - f(x'x^{-1}s^{-1}))\, d\nu(s)

and the latter integral is arbitrarily small provided

x'x^{-1}

is in a suitable neighborhood of

e

(§ 2, No. 5, Prop. 8).

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

Cited by

Nothing.

Cites

§ 2, No. 5, Prop. 8
No. 2, Prop. 5