Weak law of large numbers for iterates of random-valued functions

Given a probability space (Omega, A, P), a complete and separable metric space X with the sigma-algebra B of all its Borel subsets and a B circle times A-measurable f : X x Omega -> X we consider its iterates f(n) defined on X x Omega(N) by f(0) (x, omega) = x and f(n) (x, omega) = f (f(n-1) (x, omega), omega(n)) for n is an element of N and provide a simple criterion for the existence of a probability Borel measure pi on X such that for every x is an element of X and for every Lipschitz and bounded psi : X -> R the sequence (1/n Sigma(n-1)(k=0) psi (f(k)(x, .) ))(n is an element of N) converges in probability to integral(X) psi(y)pi(dy).