  Let $(M,\omega)$ be a K\"ahler manifold and let $K$ be a compact group that
acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on
probability measures on $M$. First of all we identify an abstract setting for
the momentum mapping and give numerical criteria for stability, semi-stability
and polystability. Next we apply this setting to the action of $K^\mathbb{C}$
on measures. We get various stability criteria for measures on K\"ahler
manifolds. The same circle of ideas gives a very general surjectivity result
for a map originally studied by Hersch and Bourguignon-Li-Yau.
