Example

— 4) One can take $f \in \mathscr{C}(X)$, $g \in \mathscr{K}(X)$ and $\mu \in \mathscr{C}'(G)$ (with $\psi = 1$).

5) If G operates properly on X, one can take $f \in \mathscr{K}(X)$, $g \in \mathscr{K}(X)$ and $\mu \in \mathscr{M}(G)$ (with $\psi = 1$).

6) One can take $f \in \mathbf{L}^p(X, \beta)$, $g \in \mathbf{L}^q(X, \beta)$ and $\mu \in \mathscr{M}^\rho(G)$, where $1 \leq p < +\infty$, $\frac{1}{p} + \frac{1}{q} = 1$, $\rho = \chi^{-1/q}$. The conditions (i) and (ii) are satisfied by Props. 5 and 6. Let us prove (iii). We have seen that $\mu$ is carried by a set S that is a countable union of compact sets. Let us take for $\psi$ the characteristic function of S. The function $h$ is $(\mu \otimes \beta)$-measurable: for, the function $(s, x) \mapsto g(x)\chi(s^{-1})\psi(s)$ is so, as is the function $(s, x) \mapsto f(s^{-1}x)$ by Lemma 1. Moreover, $g$ being zero outside a countable union of $\beta$-integrable sets, $h$ is zero outside a countable union of $(\mu \otimes \beta)$-integrable sets. We then have (Ch. V, §8, No. 3, Prop. 7):

But since $g$ (resp. $\psi$) is zero outside a countable union of integrable sets, the upper integrals of the second member of (9) are equal to the essential upper integrals (Ch. V, §1, No. 2, Prop. 7). Now (Ch. V, §5, No. 3, Prop. 3)

since $\mu = \psi \cdot \mu$. By Prop. 6, this last integral is finite and is equal to $(|\mu| *^\beta |f|)(x)$ locally $\beta$-almost everywhere. Therefore

and J is finite since $g \in L^q$ and $|\mu| *^\beta |f| \in L^p$ (Prop. 6). Therefore h is $(\mu \otimes \beta)$-integrable.

The formula (8) then means that the endomorphism $g \mapsto (\chi \cdot \check{\mu}) * g$ of $L^q(X, \beta)$ is, for $\mu \in \mathscr{M}^\rho(G)$, the transpose of the endomorphism $f \mapsto \mu * f$ of $L^p(X, \beta)$.