remarks and some examples of its application can be found in Section \[sec:examples\]. We explain the natural relation of our result with mass transportation in Section \[sec:masstransport\].

Denote by $C^\infty(P)$ the space of smooth functions on a smooth manifold $P$, and by $C^\infty_c(P)$ the compactly supported ones.

\[thm:families\] 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 family $(\mu_t)_{t{\geqslant}0}$ in $\mathscr M_0$ such that $$\mu_0=\mu\qquad \textrm{and}\qquad \left.\frac{d}{dt}\int\phi\,d\mu_t\right|_{t=0+}=\langle\eta,\phi\rangle$$ for all $\phi\in C^\infty_c(P)$. Such a family exists if, and only if, $\langle\eta,f\rangle{\geqslant}0$ for all nonnegative functions $f\in C^\infty_c(P)$ that vanish on the support of $\mu$.

If $\mu$ is a probability measure, the family $(\mu_t)_{t{\geqslant}0}$ can be realized as a family of probability measures if additionally $\langle \eta,1\rangle=0$.

In the statement of the corollary, the equation $\langle\eta,1\rangle=0$ means that, given any locally-finite partition of unity $\{\psi_i\}_i$ of $C^\infty$ nonnegative functions $\psi_i{\geqslant}0$ on $P$ such that $\sum_{i=1}^\infty\psi_i(x)=1$ for all $x\in P$, we have $$\sum_i|\langle\eta,\psi_i\rangle|<+\infty\quad\textrm{and}\quad\sum_i\langle\eta,\psi_i\rangle=0.$$

The same corollary holds mutatis mutandis for compactly supported $\mu$ and $\eta$ 