Suppose that $X \subset \mathbf{R}^n$ is a compact set, $s > 0$, and for each $\varepsilon > 0$, there exists $C_\varepsilon > 0$ and a probability measure $\mu_\varepsilon$ supported on $X$ such that $|\widehat{\mu_\varepsilon}(\xi)| \leq A_\varepsilon |\xi|^{\varepsilon - s}$. It is then true that there exists a single probability measure $\mu$ supported on $X$ such that $|\widehat{\mu}(\xi)| \leq B_\varepsilon |\xi|^{\varepsilon - s}$ for all $\varepsilon > 0$ and some other family of constants $B_\varepsilon > 0$.
I tried to solve this problem in the positive by taking weak limits of the family of measures, but I run into problems if the constants $\{ C_\varepsilon \}$ grow rapidly as $\varepsilon \to 0$, which makes me think the statement might be false, though I have no idea what a counterexample to this statement might look like.