caa a22 4500 469069082 CHVBK 20180323132916.0 cr unu---uuuuu 170328e19921001xx s 000 0 eng 10.1007/BF02808060 doi (NATIONALLICENCE)springer-10.1007/BF02808060 Maharam Dorothy Department of Mathematics, Northeastern University, 02115, Boston, MA, USA aut A "planar” representation for generalized transition kernels [Elektronische Daten] [Dorothy Maharam] In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of "Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is "sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ℝ: an ordinate set (in which vertical sections have Lebesgue measure), an "atomic” set contained inX × (−ℕ), and a "singular” set with null sections. The method incidentally produces and exploits a "reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing "uniformly σ-finite” measures (in the "standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2]. Hebrew University, 1992 Israel Journal of Mathematics Springer-Verlag 78/2-3(1992-10-01), 281-307 0021-2172 78:2-3<281 1992 78 11856 https://doi.org/10.1007/BF02808060 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02808060 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Maharam Dorothy Department of Mathematics, Northeastern University, 02115, Boston, MA, USA aut NATIONALLICENCE 773 0- Israel Journal of Mathematics Springer-Verlag 78/2-3(1992-10-01), 281-307 0021-2172 78:2-3<281 1992 78 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer