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2000 | 166 | 3 | 281-303
Tytuł artykułu

Product liftings and densities with lifting invariant and density invariant sections

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given two measure spaces equipped with liftings or densities (complete if liftings are considered) the existence of product liftings and densities with lifting invariant or density invariant sections is investigated. It is proved that if one of the marginal liftings is admissibly generated (a subclass of consistent liftings), then one can always find a product lifting which has the property that all sections determined by one of the marginal spaces are lifting invariant (Theorem 2.13). For a large class of measures Theorem 2.13 is the best possible (Theorem 4.3). When densities are considered, then one can always have a product density with measurable sections, but in the case of non-atomic complete marginal measures there exists no product density with all sections being density invariant. The results are then applied to stochastic processes.
Słowa kluczowe
Rocznik
Tom
166
Numer
3
Strony
281-303
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-12-20
poprawiono
2000-05-10
Twórcy
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, musial@math.uni.wroc.pl
autor
  • Department of Statistics, University of Piraeus, 80 Karaoli and Dimitriou street, 185 34 Piraeus, Greece , macheras@unipi.gr
Bibliografia
  • [1] P.Billingsley, Probability and Measure, Wiley, New York, 1979.
  • [2] A.Blass, handwritten notes, 1999.
  • [3] D. L.Cohn, Liftings and the construction of stochastic processes, Trans. Amer. Math. Soc. 246 (1978), 429-438.
  • [4] D. H.Fremlin, Stable sets of measurable functions, Note of 17 May 1983.
  • [5] S.Graf and H. von Weizsäcker, On the existence of lower densities in non- complete measure spaces, in: Measure Theory (Oberwolfach, 1975), A. Bellow and D. Kölzow (eds.), Lecture Notes in Math. 541, Springer, Berlin, 1976, 155-158.
  • [6] A.Ionescu and C. Tulcea, Topics in the Theory of Liftings, Springer, Berlin, 1969.
  • [7] G.Koumoullis, On perfect measures, Trans. Amer. Math. Soc., 264 (1981), 521-537.
  • [8] N. D.Macheras, K. Musiał and W. Strauss, On products of admissible liftings and densities, Z. Anal. Anwendungen 18 (1999), 651-667.
  • [9] N. D.Macheras and W. Strauss, On products of almost strong liftings, J. Austral. Math. Soc. Ser. A 60 (1996), 1-23.
  • [10] W.Sierpiński, Fonctions additives non complètement additives et fonctions non mesurables, Fund. Math. 30 (1939), 96-99.
  • [11] M.Talagrand, Pettis Integral and Measure Theory, Mem. Amer. Math. Soc. 307 (1984).
  • [12] M.Talagrand, On liftings and the regularization of stochastic processes, Probab. Theory Related Fields 78 (1988), 127-134.
  • [13] M.Talagrand, Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations, Ann. Inst. Fourier (Grenoble) 32 (1989), 39-69.
  • [14] M.Talagrand, Measurability problems for empirical processes, Ann. Probab. 15 (1987), 204-212.
  • [15] T.Traynor, An elementary proof of the lifting theorem, Pacific J. Math. 53 (1974), 267-272.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv166i3p281bwm
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