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Desintegration and perfectness of measure spaces. (English) Zbl 0669.28002

The paper consists of two parts. In the first one the author proves that each finite measure P can be uniquely represented in the form \(P=Q_ 1+Q_ 2+Q_ 3,\) where \(Q_ 1\) is compact, \(Q_ 2\) is perfect and purely non-compact (i.e. if \(Q\ll Q_ 2\) and Q is compact then \(Q=0)\) and \(Q_ 3\) is purely non-perfect. This generalizes a result of D. Ramachandran [Perfect measures. Part II (1979; Zbl 0523.60006)].
In the second part the author investigates relations between desintegration (with respect to countably generated \(\sigma\)-algebras) in the sense of J. K. Pachl [Math. Scand. 43, 157-168 (1978; Zbl 0402.28006)] and of D. Ramachandran (loc. cit.). In particular she proves that each Pachl-desintegrable with respect to countably generated \(\sigma\)-algebras probability is perfect and applies the Ramachandran- desintegration with respect to countably generated \(\sigma\)-algebras to the problem of inheritance of compactness by thick subsets.
Reviewer: K.Musiał

MSC:

28A50 Integration and disintegration of measures
28A12 Contents, measures, outer measures, capacities
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References:

[1] BELL, J. L. and SLOMSEN, A. B.: Models and Ultraproducts, 3. Aufl., North-Holland/American Elsevier Publishing Company, (1974)
[2] MAITRA, A. and RAMAKRISHNAN, S.: Factorization of Measures and Normal Conditional Distributions. To appear in Proc. Amer. Math. Soc. (1988) · Zbl 0656.60006
[3] MUSIAL, K.: Existence of Proper Conditional Probabilities. Z. Wahrscheinlichkeitstheorie verw. Geb. 22 (1972), 8-12 · Zbl 0224.60001 · doi:10.1007/BF00538901
[4] MUSIAL, K.: Inheritness of Compactness and Perfectness of Measures by Thick Subsets. Measure Theory, Oberwolfach 1975, Lecture Notes in Mathematics 541, Springer-Verlag, Berlin (1976), 39-41 · Zbl 0341.28003
[5] PACHL, J. K.: Disintegration and Compact Measures. Math. Scand. 43 (1978), 157-168 · Zbl 0402.28006
[6] RAMACHANDRAN, D.: Perfect Measures, Part I, ISI Lecture Notes, No. 5, The Macmillan Company of India Limited (1979)
[7] RAMACHANDRAN, D.: Perfect Measures, Part II, ISI Lecture Notes, No. 7, The Macmillan Company of India Limited (1979)
[8] SHOENFIELD, J. R.: Martin’s Axiom. Amer. Math. Monthly 82 (1975), 610-617 · Zbl 0314.02069 · doi:10.2307/2319691
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