Non-divergence of translates of certain algebraic measures

Let G and H ⊂ G be connected reductive real algebraic groups defined over ℚ, and admitting no nontrivial ℚ-characters. Let Γ ⊂ G(ℚ) be an arithmetic lattice in G, and π : G → Γ\G be the natural quotient map. Let μ_H denote the H-invariant probability measure on the closed orbit π(H). Suppose that π(Z(H)) is compact, where Z(H) denotes the centralizer of H in G. We prove that the set {μ_H ·g : g ∈ G} of translated measures is relatively compact in the space of all Borel probability measures on Γ\G, where μ_H ·g(E) = μ_H(Eg^-1) for all Borel sets E ⊂ Γ\G.