  We give a systematic treatment of the stability theory for action of a real
reductive Lie group G on a topological space. More precisely, we introduce an
abstract setting for actions of non-compact real reductive Lie groups on
topological spaces that admit functions similar to the Kempf-Ness function. The
point of this construction is that one can characterize stability,
semi-stability and polystability of a point by numerical criteria, that is in
terms of a function called maximal weight. We apply this setting to the actions
of a real non-compact reductive Lie group G on a real compact submanifold M of
a Kaehler manifold Z and to the action of G on measures of M.
