Stability of measures on K\"ahler manifolds

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.