A characterization of orthogonal transition kernels. (English) Zbl 0563.60006

Weis, Lutz W.

Ann. Probab. 12, 1224-1227 (1984).

Author’s summary: ”A transition kernel \(\mu =(\mu_ y)_{y\in Y}\) between Polish spaces X and Y is completely orthogonal if there is a perfect statistic \(\phi\) :X\(\to Y\) for \(\mu\), i.e. the fibers of the Borel map \(\phi\) separate the \(\mu_ y\). Equivalent properties are:
a) orthogonal, finitely additive measures p, q on Y induce orthogonal mixtures \(\mu^ p\), \(\mu^ q\) on X;
b) the Markov operator defined by \(\mu\) is surjective on a certain class of Borel functions.”
This is an elegant supplement to results of R.D. Mauldin, D. Preiss and the reviewer, Ann. Probab. 11, 970-988 (1983; Zbl 0528.60006).

Reviewer: H.von Weizsäcker

MSC:

60A10	Probabilistic measure theory
47A67	Representation theory of linear operators
62B05	Sufficient statistics and fields

Keywords:

orthogonal measures; Riesz homomorphism; finitely additive measures

Citations:

Zbl 0528.60006