Mattila defines in Fourier Analysis and Hausdorff Dimension the Fourier dimension of a set $A\subset\mathbb{R}^n$ to be
$\mathrm{dim_F}A = \sup\{s\leq n: \exists \mu \in \mathcal{M}(A):|\hat{\mu}(x)| \leq |x|^{-s/2}\forall x \in \mathbb{R}^n\}$
where $\mathcal{M}(A) = \{\mu: \text{$\mu$ is a Borel measure on $A$}, 0 < \mu(A) < \infty, \mathrm{spt}(\mu) \subset A\}$ where $\mathrm{spt}(\mu)$ is the support of $\mu$. To my knowledge Mattila doesn't mention is explicitly, but $s$ is assumed to be on the interval $(0, n]$.
My question is that knowing that there exists sets $A$ with zero Fourier dimension, how do we know that there exists such a measure $\mu$ that $|\hat{\mu}(x)| \leq 1$? Is it just as simple as taking a normlized $\mu$ with $\mu(A) = 1$ (and in particular $\int_{\mathrm{spt}(\mu)}d\mu = 1$) and then bounding
$$|\hat{\mu}(x)| = |\int_{\mathbb{R}^n}e^{2\pi i \xi \cdot x}d\mu(\xi)| = |\int_{\mathrm{spt}(\mu)}e^{2\pi i \xi \cdot x}d\mu(\xi)| \leq \int_{\mathrm{spt}(\mu)}|e^{2\pi i \xi \cdot x}|d\mu(\xi) = 1$$