If $x \in \mathbb{R}^D$, then applying
R1502: Complex-linearity of complex integral yields
\begin{equation}
\begin{split}
((\lambda f) * g) (x)
& = \int_{\mathbb{R}^D} \lambda f(y) g(x - y) \, \mu(d y) \\
& = \lambda \int_{\mathbb{R}^D} f(y) g(x - y) \, \mu(d y)
= \lambda (f * g)(x)
\end{split}
\end{equation}
Since $x \in \mathbb{R}^D$ was arbitrary, this is true for all $x \in \mathbb{R}^D$ and the proof is complete. $\square$