set of irrational numbers in $[0,1]$ has $\mu$ measure zero. In this case, Theorem \[thm:families\] and Corollary \[cor:two-sided\] show that the possible deformations are the same as for the uniform measure on $[0,1]$; see Example \[ex:uniform\]. Many of the distributions thus arising are supported in sets that are disjoint from the set holding the measure; as an example, take $\eta=\partial^k_x\delta_{\sqrt2/2}$, for any positive integer $k$.

We additionally remark that the existence of ugly examples like these is what forces us to the rather abstract treatment of the theory that we have resorted to in proving the results of Section \[sec:characterization\] rather than a more constructive and explicit technique.

\[rmk:caratheodory\] Here, we want to establish the relation between our result and the Carathéodory positivity criterion for holomorphic functions [@caratheodory]. Carathéodory proved that a necessary and sufficient condition for the signed measure $\mu$ on the interval $[0,2\pi)$ to be positive is for its Fourier coefficients $$a_n=\int_0^{2\pi}e^{-in\theta}d\mu(\theta)$$ to satisfy $$\label{eq:caratheodory}
  \sum_m\sum_na_{m-n}\lambda_m\overline{\lambda_n}{\geqslant}0$$ for any complex numbers $\lambda_0,\lambda_1,\dots,\lambda_N$.

Now consider the case in which we have a family of positive measures $(\mu_t)_{t{\geqslant}0}$ with $d\mu_t/dt|_{t=0+}=\eta$, as in Theorem \[thm:families\]. Then we have Fourier coefficients $a_n(t)=\int_0^{2\pi}e^{-in\theta}d\mu_t(\theta)$ defined for each $t{\geqslant}0$, and the Fourier coefficients of $\eta$ will 