Abstract.Given a locally compact group G acting on a locally compact space X and a probability measure on G, a real Borel function f on X is called -harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of -admissible neighbourhoods of the identity, relative to X, then every bounded -harmonic function on X is constant. Consequently, for spread out , the bounded -harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded -harmonic functions on X are constant which extends Furstenbergs result for connected semisimple Lie groups.