Example

— 1) For any $\varphi$, the measures $\varepsilon_{x_i}$, where $x_i \in X_i$ for $1 \leq i \leq n$, are always convolvable and have convolution product $\varepsilon_y$, with $y = \varphi(x_1, x_2, \ldots, x_n)$. Consequently, if each of the $\mu_i$ has finite support, then the $\mu_i$ are convolvable and $\mu_1 * \cdots * \mu_n$ has finite support. In particular, let $M$ be a monoid$^1$ equipped with a locally compact topology; if one takes $\varphi$ to be the law of composition in $M$ then the measures on $M$ with finite support form, for convolution, an algebra that is none other than the algebra of the monoid $M$ (over $\mathbf{R}$ or over $\mathbf{C}$, according as one considers real or complex measures) (A, III, §2, No. 6).

2) Let $M$ be a monoid equipped with the discrete topology; assume that for each $m \in M$, there are only finitely many pairs $(m', m'') \in M \times M$ such that $m'm'' = m$; this amounts to saying that the law of composition in $M$ is a proper mapping of $M \times M$ into $M$; the measures on $M$ then form an algebra for convolution, an algebra that is none other than the total

(1) Monoïde, in the sense of Exer. 17 of Ch. VII, §1.

algebra of the monoid $M$ (A, III, §2, No. 10); we note the following two special cases:

a) $M = N$, the law of composition being addition. To every measure $\mu$ on $N$, let us associate the formal series

in an indeterminate $t$. Then $S(\mu * \mu') = S(\mu)S(\mu')$. An analogous remark holds for formal series in any number of indeterminates.

$*b)$ $M = N^*$, the law of composition being multiplication. To every measure $\mu$ on $N^*$, let us associate the formal Dirichlet series

Then $D(\mu * \mu') = D(\mu)D(\mu')$.

3) Let $X, Y, Z$ be locally compact spaces, $\varphi$ a continuous mapping of $X \times Y$ into $Z$. If $x \in X$ and $\mu$ is a measure on $Y$, to say that $\varepsilon_x$ and $\mu$ are $\varphi$-convolvable comes to saying that the mapping $\varphi(x, \cdot)$ of $Y$ into $Z$ is $\mu$-proper. One then has $\varepsilon_x * \mu = \varphi(x, \cdot)(\mu)$.