Definition 1

*Let $X_1, \ldots, X_n$ be locally compact spaces, $\mu_i$ a measure on $X_i$ ($1 \leq i \leq n$); let X be the product of the $X_i$, $\mu$ that of the $\mu_i$. Let $\varphi$ be a mapping of X into a locally compact space Y. One says that the sequence $(\mu_i)$ is $\varphi$-convolvable, or that $\mu_1, \ldots, \mu_n$ are $\varphi$-convolvable, if $\varphi$ is $\mu$-proper; in this case, the image $\nu = \varphi(\mu)$ of $\mu$ under $\varphi$ is called the convolution product of the $\mu_i$ for $\varphi$, and is denoted $*_{\varphi}(\mu_i)_{1 \leq i \leq n}$, or $*_{i=1}^n \mu_i$, or $\mu_1 * \mu_2 * \cdots * \mu_n$.

The last two notations are of course used only when there can be no doubt as to $\varphi$.

Let f be a function on Y, with values in a Banach space or in $\overline{\mathbf{R}}$. In order that f be essentially integrable for $\mu_1 * \cdots * \mu_n$, it is necessary and sufficient that the function

be essentially integrable for $\mu_1 \otimes \mu_2 \otimes \cdots \otimes \mu_n$, in which case

a formula that may be regarded as defining $\mu_1 * \cdots * \mu_n$ when one takes $f \in \mathscr{K}(Y)$.

The definitions imply at once that the $\mu_i$ are convolvable if and only if the $|\mu_i|$ are. When this is the case,

(Ch. VI, §2, No. 10), that is,

If the $\mu_i$ are convolvable and positive, and if $\nu_i$ is a measure on $X_i$ such that $0 \leq \nu_i \leq \mu_i$, then the $\nu_i$ are convolvable and

Suppose $\mu_1, \mu_2, \ldots, \mu_n$ are convolvable, and that $\mu'_1, \mu_2, \ldots, \mu_n$ are convolvable ($\mu'_1$ being a measure on $X_1$). By Ch. V, §6, No. 3, Cor. 1 of Prop. 6, $\mu_1 + \mu'_1, \mu_2, \ldots, \mu_n$ are convolvable and