On uniqueness of invariant means

The following results on uniqueness of invariant means are shown: (i) Let G be a connected almost simple algebraic group defined over Q. Assume that G(R), the group of the real points in G, is not compact. Let p be a prime, and let G(Z(p)) be the compact p-adic Lie group of the Z(p)-points in G. Then the normalized Haar measure on G(Z(p)) is the unique invariant mean on L-infinity(G(Z(p))). (ii) Let G be a semisimple Lie group with finite centre and without compact factors, and let Gamma be a lattice in G. Then integration against the G-invariant probability measure on the homogeneous space G/Gamma is the unique Gamma-invariant mean on L-infinity(G/Gamma).