family $(\mu_t)_{t\in{\mathbb{R}}}$ can be realized as a family of probability measures if additionally $\langle\eta,1\rangle=0$.

\[ex:two-sided-delta\] Going back to the situation of Examples \[ex:deformationsdelta\] and \[ex:probdeformationsdelta\], we now determine the possible two-sided deformations of $\delta_0$ both as two-sided families of positive, Radon measures and as two-sided families of probability distributions on $P$. From Corollary \[cor:two-sided\], we see that the condition that needs to be satisfied by the derivative $\eta$ is that $\langle\eta,f\rangle=0$ for all $f\in C^\infty_c({\mathbb{R}})$ such that $f{\geqslant}0$ throughout ${\mathbb{R}}$ and $f(0)=0$. In this case, the quadratic term of the Taylor expansion of $f$ impedes $\eta$ from having a term of order 2, so for the case of arbitrary positive measures we get that $\eta$ must be of the form $$\eta=\nu+a\,\partial_x\delta_0,$$ for a positive measure $\nu$ and a real number $a$. Similarly, in the case of families of probability measures the additional condition that $\langle\eta,1\rangle=0$ forces $\nu$ to vanish and $\eta$ to be of the form $$\eta=a\,\partial_x\,\delta_0,\quad a\in{\mathbb{R}}.$$

\[ex:uniform\] *Distributions of any degree may arise as derivatives of families of measures.* Consider the case in which $\mu$ is the uniform probability measure on a bounded, open set $U\subseteq P$. Then any distribution $\eta$ supported in $\overline U$ satisfies the condition 