deformed within the space of probability measures. In this case, we have the additional requirement that the derivative $\eta$ satisfy $\langle\eta,1\rangle=0$. The reader will immediately recognize that this amounts to the requirement that the measure $\nu$ vanish everywhere or, in other words, that $\eta$ be of the form $$\eta=a\,\partial_x\delta_0+b\,\partial_x^2\delta_0$$ for $a\in {\mathbb{R}}$, $b>0$.

Although the statement of Theorem \[thm:families\] deals only with one-sided families $(\mu_t)_{t{\geqslant}0}$, it is clear what the requirement is on the distribution $\eta$ for the existence of a two-sided family $(\mu_t)_{t\in{\mathbb{R}}}$ with derivative $\eta$: $\eta$ and $-\eta$ must satisfy the conditions in Theorem \[thm:families\]. For the reader’s convenience, we state this in full:

\[cor:two-sided\] Let $\mathscr M_0$ be the set of positive, Radon measures on a $C^\infty$ manifold $P$. Let $\mu$ be an element of $\mathscr M_0$ and let $\eta$ be an element of the space $\mathscr D'$ of distributions on $P$. We look for conditions for the existence of a (two-sided) family $(\mu_t)_{t\in {\mathbb{R}}}$ in $\mathscr M_0$ such that $\mu_0=\mu$ and $\frac{d}{dt}\int \phi\,d\mu_t|_{t=0}=\langle\eta,\phi\rangle$ for all $\phi\in C^\infty_c(P)$. Such a family exists if, and only if, $\langle\eta,f\rangle=0$ for all nonnegative functions $f\in C^\infty_c(P)$ that vanish in the support of $\mu$.

If $\mu$ is a probability measure, the 