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

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

Zbl 0528.60006
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