The convolution equation of Choquet and Deny on [IN]-groups

Let σ be a probability measure on a locally compact groupG. A real Borel functionf onG is called σ-harmonic if it satisfies the convolution equation σ*f=f. Given that σ isnonsingular with its translates, we show that the bounded σ-harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If σ is nondegenerate and absolutely continuous, we solve the more general equation σ*μ=μ for positive measure μ on those groups which are metrizable and separable.