Let $M = (\mathbb{R}^D, \mathcal{B}(\mathbb{R}^D))$ be a D2763: Euclidean real Borel measurable space such that

(i)	$\mu : \mathcal{B}(\mathbb{R}^D) \to [0, \infty]$ is an D85: Unsigned basic measure on $M$
(ii)	\begin{equation} \mu(\mathbb{R}^D) < \infty \end{equation}

Fix $\xi \in \mathbb{R}^D$. Applying results

(i)	R2533: Triangle inequality for complex integral
(ii)	R2749: Planar wave kernel is constant in Euclidean norm

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} $\square$