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Factorization of probability measures and absolutely measurable sets. (English) Zbl 0554.60001

The main result of the paper is as follows.
Theorem. Let Y be a separable metric space. Then the following conditions on Y are equivalent.
(a) Y is absolutely measurable, i.e., if \(\tilde Y\) is a metric completion of Y and \(\lambda\) is a probability measure on the Borel \(\sigma\)-field of \(\tilde Y,\) then Y is \(\lambda\)-measurable.
(b) For any measurable space (X,\({\mathcal A})\) and any probability measure P on (X\(\times Y,{\mathcal A}\times {\mathcal B}(Y))\), P can be factored.
(c) For any Polish space X and any probability measure P on (X\(\times Y,{\mathcal B}(X)\times {\mathcal B}(Y))\), P can be factored.

MSC:

60A10 Probabilistic measure theory
60A05 Axioms; other general questions in probability
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A50 Integration and disintegration of measures
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References:

[1] Miloslav Jiřina, Conditional probabilities on strictly separable \?-algebras, Czechoslovak Math. J. 4(79) (1954), 372 – 380 (Russian, with English summary). · Zbl 0058.11801
[2] Топология. Том 2, Транслатед фром тхе Енглиш бы М. Ја. Антоновский, Издат. ”Мир”, Мосцощ, 1969 (Руссиан).
[3] E. Marczewski, On compact measures, Fund. Math. 40 (1953), 113 – 124. · Zbl 0052.04902
[4] Jan K. Pachl, Disintegration and compact measures, Math. Scand. 43 (1978/79), no. 1, 157 – 168. · Zbl 0402.28006
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