Abstract
Let \({\text{(}}\Omega {\text{,}}A,,P)\) be a probability space and Π a partition of Ω. A necessary and sufficient condition is given for the existence of a σ-additive and measurable disintegration of P on Π. It is also shown that P admits a σ-additive (but not measurable) disintegration on Π whenever \({\text{(}}\Omega {\text{,}}A,,)\) is a standard space and the set (ω1, ω2):ω1 and ω2 are in the same element of Π} is coanalytic in Ω×Ω. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using σ-additive disintegrations as conditional probabilities.
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Berti, P., Rigo, P. Sufficient Conditions for the Existence of Disintegrations. Journal of Theoretical Probability 12, 75–86 (1999). https://doi.org/10.1023/A:1021792409934
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DOI: https://doi.org/10.1023/A:1021792409934