  Let (G, X) be a transformation group where the group $G$ does not necessarily
act freely on the space X. We investigate the extent to which the action of G
may fail to be proper. Stability subgroups are used to define new notions of
strength of convergence in the orbit space and of measure accumulation along
orbits. By using the representation theory of the associated crossed product
C^*-algebra, we show that these notions are equivalent under certain
conditions.
