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 Zähle 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.
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When this work was completed, I became aware of the paper [6] on a related subject, written independently of the present one and finished at about the same time. Though some overlap is inevitable (in particular, versions of our Theorem 5.1 appear in both papers), the approach and emphasis are different, and the interested reader may want to study both papers in parallel.
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Kallenberg, O. Invariant Palm and related disintegrations via skew factorization. Probab. Theory Relat. Fields 149, 279–301 (2011). https://doi.org/10.1007/s00440-009-0254-2
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DOI: https://doi.org/10.1007/s00440-009-0254-2
Keywords
- Invariant disintegration
- Orbit selection
- Inversion kernel
- Skew factorization
- Duality
- Stationary random measures
- Palm kernel
Mathematics Subject Classification (2000)
- 28C10
- 60G57
- 60G10