Factorization of measures and normal conditional distributions
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- by A. Maitra and S. Ramakrishnan PDF
- Proc. Amer. Math. Soc. 103 (1988), 1259-1267 Request permission
Abstract:
Let $(Y,\mathcal {C},Q)$ be a probability space. If every probability measure $R$ on ${\mathcal {B}^1} \otimes \mathcal {C}$ with marginal $Q$ on $Y$ admits a factorization, where ${\mathcal {B}^1}$ is the Borel $\sigma$-field on the real line, $Q$ must be perfect. Conversely if $Q$ is perfect and $\mathcal {C}$ is ${\aleph _1}$-generated, then (a) for any measure $R$ on $\mathcal {A} \otimes \mathcal {C}$ with marginal $Q$, where $\mathcal {A}$ is any $\sigma$-field of subsets of a set $X$, there is a factorization; (b) for every tail-like sub-$\sigma$-field $\mathcal {D}$ of $\mathcal {C}$, there is a normal conditional distribution given $\mathcal {D}$. In special cases of interest, normal conditional distributions, satisfying additional desirable properties, are shown to exist.References
- Wolfgang Adamski, Factorization of measures and perfection, Proc. Amer. Math. Soc. 97 (1986), no. 1, 30–32. MR 831381, DOI 10.1090/S0002-9939-1986-0831381-5
- David Blackwell and Lester E. Dubins, On existence and non-existence of proper, regular, conditional distributions, Ann. Probability 3 (1975), no. 5, 741–752. MR 400320, DOI 10.1214/aop/1176996261
- David Blackwell and Ashok Maitra, Factorization of probability measures and absolutely measurable sets, Proc. Amer. Math. Soc. 92 (1984), no. 2, 251–254. MR 754713, DOI 10.1090/S0002-9939-1984-0754713-3
- D. Blackwell and C. Ryll-Nardzewski, Non-existence of everywhere proper conditional distributions, Ann. Math. Statist. 34 (1963), 223–225. MR 148097, DOI 10.1214/aoms/1177704259
- Lester E. Dubins, On conditional distributions for stochastic processes, Papers from the “Open House for Probabilists” (Mat. Inst., Aarhus Univ., Aarhus, 1971) Various Publ. Ser., No. 21, Mat. Inst., Aarhus Univ., Aarhus, 1972, pp. 72–85. MR 0400321
- Lester E. Dubins, Finitely additive conditional probabilities, conglomerability and disintegrations, Ann. Probability 3 (1975), 89–99. MR 358891, DOI 10.1214/aop/1176996451
- Lester E. Dubins, Measurable tail disintegrations of the Haar integral are purely finitely additive, Proc. Amer. Math. Soc. 62 (1976), no. 1, 34–36 (1977). MR 425071, DOI 10.1090/S0002-9939-1977-0425071-5
- Lester E. Dubins and David Heath, With respect to tail sigma fields, standard measures possess measurable disintegrations, Proc. Amer. Math. Soc. 88 (1983), no. 3, 416–418. MR 699405, DOI 10.1090/S0002-9939-1983-0699405-3
- David Heath and William Sudderth, On finitely additive priors, coherence, and extended admissibility, Ann. Statist. 6 (1978), no. 2, 333–345. MR 464450
- David A. Lane and William D. Sudderth, Diffuse models for sampling and predictive inference, Ann. Statist. 6 (1978), no. 6, 1318–1336. MR 523766
- E. Marczewski, On compact measures, Fund. Math. 40 (1953), 113–124. MR 59994, DOI 10.4064/fm-40-1-113-124
- Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
- Kazimierz Musiał, Inheritness of compactness and perfectness of measures by thick subsets, Measure theory (Proc. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 541, Springer, Berlin, 1976, pp. 31–42. MR 0442181
- Jan K. Pachl, Disintegration and compact measures, Math. Scand. 43 (1978/79), no. 1, 157–168. MR 523833, DOI 10.7146/math.scand.a-11771
- Roger A. Purves and William D. Sudderth, Some finitely additive probability, Ann. Probability 4 (1976), no. 2, 259–276. MR 402888, DOI 10.1214/aop/1176996133
- C. Ryll-Nardzewski, On quasi-compact measures, Fund. Math. 40 (1953), 125–130. MR 59997, DOI 10.4064/fm-40-1-125-130 V. V. Sazonov, On perfect measures, Amer. Math. Soc. Transl. (2) 48 (1965), 229-254. R. Sikorski, Boolean algebras, 2nd edition, Springer-Verlag, Berlin and New York, 1964.
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 1259-1267
- MSC: Primary 60A10; Secondary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0955019-3
- MathSciNet review: 955019