Proposition 10

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

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

Let $f, g$ be in $\mathscr{L}(G)$ . Assume that one of these two functions is continuous or is zero on the complement of a countable union of compact sets. If $f$ and $g$ are convolvable, then the function $f * g$ is given locally $\beta$ -almost everywhere by

(f * g)(x) = \int_G g(s^{-1}x)f(s)\chi(s^{-1})\, d\beta(s) = \int_G f(xs^{-1})g(s)\chi'(s^{-1})\, d\beta(s).

This follows from Prop. 2 of No. 1, and the remarks in No. 4.

In particular, if $f * g$ is continuous and is given at every point by (15), then

(f * g)(e) = \int g(s^{-1})f(s)\chi(s^{-1})\, d\beta(s) = \int f(s^{-1})g(s)\chi'(s^{-1})\, d\beta(s).

Still more particularly, if $\beta$ is a left and right Haar measure, and if $f * g$ and $g * f$ are continuous and are given at every point by (15) and the analogous formula for $g * f$ , then

(f * g)(e) = (g * f)(e) = \int f(s)g(s^{-1})\, d\beta(s).

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

Cited by

Nothing.

Cites

Prop. 2