Fix $\xi \in \mathbb{R}^D$. Applying results
we have
\begin{equation}
\begin{split}
|\mathfrak{F}_{\mu}(\xi)| = \left| \int_{\mathbb{R}^D} C e^{i c x \cdot \xi} \, \mu(d x) \right| & \leq \int_{\mathbb{R}^D} \left| C e^{i c x \cdot \xi} \right| \, \mu(d x) \\
& = |C| \int_{\mathbb{R}^D} \, d \mu(x)
= |C| \mu(\mathbb{R}^D)
\end{split}
\end{equation}
$\square$
