On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let $\mathbb{G}$ be a connected almost simple algebraic group defined over $\mathbb{Q}$. Assume that $\mathbb{G}(\mathbb{R})$, the group of the real points in $\mathbb{G}$, is not compact. Let $p$ be a prime, and let $\mathbb{G}({\mathbb{Z}}_{p})$ be the compact $p$-adic Lie group of the ${\mathbb{Z}}_{p}$-points in $\mathbb{G}$. Then the normalized Haar measure on $\mathbb{G}({\mathbb{Z}}_{p})$ is the unique invariant mean on $L^{\infty }(\mathbb{G}({\mathbb{Z}}_{p}))$.

(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 $L^{\infty } (G/\Gamma )$.