Distribution preserving transformations of sequences on compact metric spaces

Let X be ii compact metric space. For s is an element of X let mu(x) denote the point measure in x, i.e. mu(x)(B) = 1 if x is an element of B and mu(B) = 0 else for each Borel set B subset of or equal to X. For each sequence x = (x(n))(n is an element of N) consider the Borel measures mu(x,N) = 1/N Sigma(n=1)(N) mu(xn). Let the weak topology on the space of all Borel probability measures be induced by the metric d. Call two sequences x and x' equivalent (x similar to x') if lim(N-infinity) d(mu(x,N), mu(x',N)) = 0. Let furthermore Y be another compact metric space, Consider sequences f = (f(n))(n is an element of N) Of maps f(n) : X--> Y and the induced sequences f(x) = (f(n)(X-n))(n is an element of N). The object of this paper is to characterize all f such that x similar to x' always implies f(x) similar to f(x'). The topic is closely related to more elementary questions on Cesaro means.