A note on integral representation of Feller kernels

R. Rębowski

Annales Polonici Mathematici (1991)

  • Volume: 56, Issue: 1, page 93-96
  • ISSN: 0066-2216

Abstract

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We consider integral representations of Feller probability kernels from a Tikhonov space X into a Hausdorff space Y by continuous functions from X into Y. From the existence of such a representation for every kernel it follows that the space X has to be 0-dimensional. Moreover, both types of representations coincide in the metrizable case when in addition X is compact and Y is complete. It is also proved that the representation of a single kernel is equivalent to the existence of some non-direct product measure on the product space Y ℕ .

How to cite

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R. Rębowski. "A note on integral representation of Feller kernels." Annales Polonici Mathematici 56.1 (1991): 93-96. <http://eudml.org/doc/262416>.

@article{R1991,
abstract = {We consider integral representations of Feller probability kernels from a Tikhonov space X into a Hausdorff space Y by continuous functions from X into Y. From the existence of such a representation for every kernel it follows that the space X has to be 0-dimensional. Moreover, both types of representations coincide in the metrizable case when in addition X is compact and Y is complete. It is also proved that the representation of a single kernel is equivalent to the existence of some non-direct product measure on the product space $Y^ℕ$.},
author = {R. Rębowski},
journal = {Annales Polonici Mathematici},
keywords = {Feller kernel; integral representation; integral representations of Feller probability kernels; Tikhonov space; product measure},
language = {eng},
number = {1},
pages = {93-96},
title = {A note on integral representation of Feller kernels},
url = {http://eudml.org/doc/262416},
volume = {56},
year = {1991},
}

TY - JOUR
AU - R. Rębowski
TI - A note on integral representation of Feller kernels
JO - Annales Polonici Mathematici
PY - 1991
VL - 56
IS - 1
SP - 93
EP - 96
AB - We consider integral representations of Feller probability kernels from a Tikhonov space X into a Hausdorff space Y by continuous functions from X into Y. From the existence of such a representation for every kernel it follows that the space X has to be 0-dimensional. Moreover, both types of representations coincide in the metrizable case when in addition X is compact and Y is complete. It is also proved that the representation of a single kernel is equivalent to the existence of some non-direct product measure on the product space $Y^ℕ$.
LA - eng
KW - Feller kernel; integral representation; integral representations of Feller probability kernels; Tikhonov space; product measure
UR - http://eudml.org/doc/262416
ER -

References

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  1. [1] R. M. Blumenthal and H. H. Corson, On continuous collections of measures, Ann. Inst. Fourier (Grenoble) 20 (2) (1970), 193-199. Zbl0195.06102
  2. [2] R. M. Blumenthal and H. H. Corson, On continuous collections of measures, in: Proc. of the Sixth Berkeley Sympos. on Math. Statistics and Probability, Vol. II, Berkeley and Los Angeles, Univ. of Calif. Press, 1972, 33-40. Zbl0253.60063
  3. [3] N. Ghoussoub, An integral representation of randomized probabilities and its applications, in: Séminaire de Probabilités XVI, Lecture Notes in Math. 920, Springer, Berlin 1982, 519-543. Zbl0493.60005
  4. [4] A. Iwanik, Integral representations of stochastic kernels, in: Aspects of Positivity in Functional Analysis, R. Nagel, U. Schlotterbeck and M. P. H. Wolff (eds.), Elsevier, 1986, 223-230. 
  5. [5] Y. Kifer, Ergodic Theory of Random Transformations, Progr. Probab. Statist. 10, Birkhäuser, Boston 1986. Zbl0604.28014

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