Definition 1

If $\mu$ and $f \cdot \beta$ are convolvable, $\mu$ and $f$ are said to be convolvable relative to $\beta$. Every density of $\mu * (f \cdot \beta)$ with respect to $\beta$ is called a convolution product of $\mu$ and $f$ relative to $\beta$ and is denoted $\mu *_{\beta} f$.

One omits $\beta$ when no confusion is possible. Convolution for several measures on $G$ and a function on $X$ is defined in an analogous manner.

The various convolution products of $\mu$ and $f$ are equal locally $\beta$-almost everywhere. If $\beta$ has support $X$ and if there exists a convolution product of $\mu$ and $f$ that is continuous, then the latter is uniquely determined; it is then called the convolution product of $\mu$ and $f$ relative to $\beta$.

Let $s \in G$ and let $f$ be a locally $\beta$-integrable complex function on $X$. Then $\varepsilon_s$ and $f$ are convolvable, and

therefore

locally $\beta$-almost everywhere.