Fix $\xi \in \mathbb{R}^N$. As a consequence of R3020: Complex Lebesgue integral under scaling, we have \begin{equation} \begin{split} \mathfrak{F}_{x \mapsto f(x / a)} (\xi) & = \int_{\mathbb{R}^N} f(x / a) e^{- 2 \pi i x \cdot \xi} \, \ell(d x) \\ & = \int_{\mathbb{R}^N} f(x / a) e^{- 2 \pi i (x / a) \cdot (a \xi)} \, \ell(d x) \\ & = |a|^N \int_{\mathbb{R}^N} f(x) e^{- 2 \pi i x \cdot (a \xi)} \, \ell(d x) \\ & = |a|^N \mathfrak{F}_f (a \xi) \end{split} \end{equation} This establishes the second claim. The second claim is a corollary to this which one obtains by applying the above procedure for a constant $b : = 1 / a$. The cases for $\mathfrak{G}$ are proved in exactly the same manner. $\square$