A Markov operator acting on
the space of continuous functions on a compact Hausdorff space which is
uniformly stable in the mean allows a topological ergodic decomposition. A
partial converse to this is obtained; if the operator has a decomposition it
is then uniformly stable in the mean when restricted to the conservative
set. The characterization of uniformly mean stable operators in terms of its
invariant structures is the major result. The problem of characterizing the
manifolds which can be the invariant manifold for some Markov operator is also
considered.
