Standard filling of a product space

T. S. Kudryk; W. E. Lyantse

Sometimes it is useful to substitute an infinite mathematical structure by a hyperfinite one. But for a hyperfinite set the conception of standardness (in the usual sense) is not defined. This disadvantage can be removed by a suitable standard filling construction. The last conception was introduced in [7] (see also [9] or [10]). It is based on some easy $Q$-$\Pi$-formalism. Here we propose a standard filling for a hyperfinite product of a family of finite probability spaces. This gives a way to replace an infinite stochastic process by a hyperfinite one.

1. Березанский Ю. М. Самосопряженные операторы в пространствах функций бесконечного числа переменных. – Киев, Наукова думка, 1978.

2. Березанский Ю. М., Кондратьев Ю. Г. Спектральные методы в бесконечномерном анализе. – Киев, Наукова думка, 1988.

3. Cartier P., Feneyrol-Perrin Y. Methods Infinitesimales en Analyse et Calcul des Probabilites. Lect. Notes in Math., 1998.

4. Davis M. Applied nonstandard analysis. A Wiley-Interscience publication, New York-London-Sydney-Toronto, 1977.

5. Diener F., Reeb G. Analyse Nonstandard. Hermann, Ed. Sc. Arts,,1989.

6. Kusraev A. G., Kutateladze S. S. Nonstandard methods of analysis. Novosibirsk, Nauka, 1990.

7. Kudryk T., Lyantse W., Chuiko G. Nearstandardness on a finite set. Matematychni Studii 3 (1993), 25–34.

8. Lutz R., Goze M. Nonstandard Analysis: a practical guide with applications. Lect. Notes in Math., 881, 1981.

9. Lyantse W. Nearstandardness on a finite set. Dissertationes Mathematicae, CCCLXIX, Warszawa, 1997.

10. Lyantse W., Kudryk T. Introduction to Nonstandard Analysis. Mathematical Studies, Monograph Series, Vol. 3, VNTL Publishers, Lviv, 1997.

11. Nelson E. Internal Set Theory: a new approach to Nonstandard Analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198.

12. Nelson E. Radically elementary probability theory. Princeton University Press, Princeton, New Jersey, 1987.

	Standard filling of a product space
Author	T. S. Kudryk, W. E. Lyantse wlanc@litech.lviv.ua, tkudryk@franko.lviv.ua Faculty of Mechanics and Mathematics, Lviv National University, 1, Universytetska str., Lviv, 79000, Ukraine, ,
Abstract	Sometimes it is useful to substitute an infinite mathematical structure by a hyperfinite one. But for a hyperfinite set the conception of standardness (in the usual sense) is not defined. This disadvantage can be removed by a suitable standard filling construction. The last conception was introduced in [7] (see also [9] or [10]). It is based on some easy $Q$-$\Pi$-formalism. Here we propose a standard filling for a hyperfinite product of a family of finite probability spaces. This gives a way to replace an infinite stochastic process by a hyperfinite one.
Keywords	hyperfinite sets, standard filling construction, nonstandard analysis, hyperfinite products, finite probability spaces, infinite stochastic processes
DOI	doi:10.30970/ms.16.2.169-184
Reference	1. Березанский Ю. М. Самосопряженные операторы в пространствах функций бесконечного числа переменных. – Киев, Наукова думка, 1978. 2. Березанский Ю. М., Кондратьев Ю. Г. Спектральные методы в бесконечномерном анализе. – Киев, Наукова думка, 1988. 3. Cartier P., Feneyrol-Perrin Y. Methods Infinitesimales en Analyse et Calcul des Probabilites. Lect. Notes in Math., 1998. 4. Davis M. Applied nonstandard analysis. A Wiley-Interscience publication, New York-London-Sydney-Toronto, 1977. 5. Diener F., Reeb G. Analyse Nonstandard. Hermann, Ed. Sc. Arts,,1989. 6. Kusraev A. G., Kutateladze S. S. Nonstandard methods of analysis. Novosibirsk, Nauka, 1990. 7. Kudryk T., Lyantse W., Chuiko G. Nearstandardness on a finite set. Matematychni Studii 3 (1993), 25–34. 8. Lutz R., Goze M. Nonstandard Analysis: a practical guide with applications. Lect. Notes in Math., 881, 1981. 9. Lyantse W. Nearstandardness on a finite set. Dissertationes Mathematicae, CCCLXIX, Warszawa, 1997. 10. Lyantse W., Kudryk T. Introduction to Nonstandard Analysis. Mathematical Studies, Monograph Series, Vol. 3, VNTL Publishers, Lviv, 1997. 11. Nelson E. Internal Set Theory: a new approach to Nonstandard Analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. 12. Nelson E. Radically elementary probability theory. Princeton University Press, Princeton, New Jersey, 1987.
Pages	169-184
Volume	16
Issue	2
Year	2001
Journal	Matematychni Studii
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