Product liftings and densities with lifting invariant and density invariant sections

Kazimierz Musiał; W. Strauss; N. Macheras

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 3, page 281-303
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Musiał, Kazimierz, Strauss, W., and Macheras, N.. "Product liftings and densities with lifting invariant and density invariant sections." Fundamenta Mathematicae 166.3 (2000): 281-303. <http://eudml.org/doc/212482>.

@article{Musiał2000,
abstract = {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.},
author = {Musiał, Kazimierz, Strauss, W., Macheras, N.},
journal = {Fundamenta Mathematicae},
keywords = {product liftings; densities; density invariant sections; product density},
language = {eng},
number = {3},
pages = {281-303},
title = {Product liftings and densities with lifting invariant and density invariant sections},
url = {http://eudml.org/doc/212482},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Musiał, Kazimierz
AU - Strauss, W.
AU - Macheras, N.
TI - Product liftings and densities with lifting invariant and density invariant sections
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 3
SP - 281
EP - 303
AB - 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.
LA - eng
KW - product liftings; densities; density invariant sections; product density
UR - http://eudml.org/doc/212482
ER -

References

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  1. [1] P.Billingsley, Probability and Measure, Wiley, New York, 1979. Zbl0411.60001
  2. [2] A.Blass, handwritten notes, 1999. 
  3. [3] D. L.Cohn, Liftings and the construction of stochastic processes, Trans. Amer. Math. Soc. 246 (1978), 429-438. Zbl0407.46039
  4. [4] D. H.Fremlin, Stable sets of measurable functions, Note of 17 May 1983. 
  5. [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. [6] A.Ionescu and C. Tulcea, Topics in the Theory of Liftings, Springer, Berlin, 1969. 
  7. [7] G.Koumoullis, On perfect measures, Trans. Amer. Math. Soc., 264 (1981), 521-537. Zbl0469.28010
  8. [8] N. D.Macheras, K. Musiał and W. Strauss, On products of admissible liftings and densities, Z. Anal. Anwendungen 18 (1999), 651-667. Zbl0947.28005
  9. [9] N. D.Macheras and W. Strauss, On products of almost strong liftings, J. Austral. Math. Soc. Ser. A 60 (1996), 1-23. Zbl0879.28008
  10. [10] W.Sierpiński, Fonctions additives non complètement additives et fonctions non mesurables, Fund. Math. 30 (1939), 96-99. Zbl0018.11403
  11. [11] M.Talagrand, Pettis Integral and Measure Theory, Mem. Amer. Math. Soc. 307 (1984). Zbl0582.46049
  12. [12] M.Talagrand, On liftings and the regularization of stochastic processes, Probab. Theory Related Fields 78 (1988), 127-134. Zbl0627.60046
  13. [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. [14] M.Talagrand, Measurability problems for empirical processes, Ann. Probab. 15 (1987), 204-212. Zbl0622.60040
  15. [15] T.Traynor, An elementary proof of the lifting theorem, Pacific J. Math. 53 (1974), 267-272. Zbl0258.46043

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