On finite invariant measures for sets of Markov operators

Abstract:

A. Brunel [1] proved that a Markovian operator P has an invariant measure if and only if each convex combination of iterates $\sum \nolimits _{n = 0}^\infty {{\alpha _n}{P^n}}$ is conservative. In the present paper this result is generalized for any commutative semigroup of Markovian operators: Let II be a semigroup; there exists a common invariant measure for II if and only if each convex combination $\sum \nolimits _{n = 1}^\infty {{\alpha _n}{P_n}}$ where $\{ {P_n}\} \subset \Pi$, is conservative.