Ergodic sequences of probability measures on commutative hypergroups

We study conditions on a sequence of probability measures {µ n } n on a commutative hyper-group K, which ensure that, for any representation π of K on a Hilbert space Ᏼ π and for any ξ ∈ Ᏼ π , (K π x (ξ)dµ n (x)) n converges to a π-invariant member of Ᏼ π. 1. Introduction. The mean ergodic theorem was originally formulated by von Neu-mann [13] for one-parameter unitary groups in Hilbert space. Riesz [11] and Yosida [16] gave the first simple proofs that if T is a bounded linear operator on a reflexive Banach space with subunitary norm, the corresponding sequence of averages (A n) n , A n = n