The Unique stationary boundary property

Let G be a locally compact group and μ an admissible probability measure on G. Let (B,ν) be the universal topological Poisson μ-boundary of (G,μ) and Πₛ(G) the universal minimal strongly proximal G-flow. This note is inspired by a recent result of Hartman and Kalantar. We show that for a locally compact second countable group G the following conditions are equivalent: (i) the measure ν is the unique μ-stationary measure on B, (ii) (B,G) (Πₛ(G),G).