  Let G be an algebraic group over a complete separable valued field k. We
discuss the dynamics of the G-action on spaces of probability measures on
algebraic G-varieties. We show that the stabilizers of measures are almost
algebraic and the orbits are separated by open invariant sets. We discuss
various applications, including existence results for algebraic representations
of amenable ergodic actions. The latter provides an essential technical step in
the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to
Bader and Furman.
