Invariant Palm and related disintegrations via skew factorization

Abstract

Let G be a measurable group with Haar measure λ, acting properly on a space S and measurably on a space T. Then any σ-finite, jointly invariant measure M on S × T admits a disintegration \({\nu \otimes \mu}\) into an invariant measure ν on S and an invariant kernel μ from S to T. Here we construct ν and μ by a general skew factorization, which extends an approach by Rother and Zahle for homogeneous spaces S over G. This leads to easy extensions of some classical propositions for invariant disintegration, previously known in the homogeneous case. The results are applied to the Palm measures of jointly stationary pairs (ξ, η), where ξ is a random measure on S and η is a random element in T.