On the trivializability of rank-one cocycles with an invariant field of projective measures

Let G be SO°(n, 1) for n⩾ 3 and consider a lattice Γ < G . Given a standard Borel probability Γ -space (Ω , μ) , consider a measurable cocycle σ: Γ × Ω → H(κ) , where H is a connected algebraic κ -group over a local field κ . Under the assumption of compatibility between G and the pair (H, κ) , we show that if σ admits an equivariant field of probability measures on a suitable projective space, then σ is trivializable. An analogous result holds in the complex hyperbolic case.