Remark 2

— Let $X$ be a locally compact space on which a locally compact group $G$ operates continuously on the right by $(x, s) \mapsto xs$. Let $\beta$ be a positive measure on $X$. Let $\chi$ be a function $> 0$ on $G \times X$, measurable for every measure on $G \times X$, such that $\delta(s)\beta = \chi(s, \cdot) \cdot \beta$ for every $s \in G$. Let $f$ be a locally $\beta$-integrable function on $X$ and let $\mu$ be a measure on $G$. If $f \cdot \beta$ and $\mu$ are convolvable (for the mapping $(x, s) \mapsto xs$ of $X \times G$ into $X$), then $(f \cdot \beta) * \mu$ has base $\beta$. We then say that $f$ and $\mu$ are convolvable relative to $\beta$; every density of $(f \cdot \beta) * \mu$ with respect to $\beta$ is called a convolution product of $f$ and $\mu$ relative to $\beta$, and is denoted $f *^\beta \mu$ or simply $f * \mu$.

Let $G^0$ be the group opposite to $G$. By $(s, x) \mapsto xs$, $G^0$ operates continuously on the left in $X$. To say that $f$ and $\mu$ are convolvable in the foregoing sense is equivalent to saying that $\mu$ and $f$ are convolvable for $G^0$ operating on the left in $X$; and the convolution products $f *^\beta \mu$ are none other than the convolution products $\mu *^\beta f$ for $G^0$ operating on the left in $X$. On the other hand, for $s \in G^0$ one has $\gamma(s)\beta = \chi(s^{-1}, \cdot) \cdot \beta$. The results of Nos. 1 and 2 may then be translated immediately into results concerning the products $f *^\beta \mu$. In particular:

1) If $s \in G$ and $f$ is locally $\beta$-integrable, then $f$ and $\varepsilon_s$ are convolvable and

2) If $f$ and $\mu$ are convolvable and if one of the conditions (i), (ii), (iii) of Prop. 2 is fulfilled, then $f *^\beta \mu$ is given locally $\beta$-almost everywhere by

We leave to the reader the task of translating the other statements. Note that if $\chi$ is continuous and $\beta$ has support $X$, then