A structural approach to solving the 6th Hilbert problem

Authors:
Yu. I. Petunin and D. A. Klyushin

Translated by:
V. Zayats

Journal:
Theor. Probability and Math. Statist. **71** (2005), 165-179

MSC (2000):
Primary 60A05

DOI:
https://doi.org/10.1090/S0094-9000-05-00656-3

Published electronically:
December 30, 2005

MathSciNet review:
2144329

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rational- and real-valued).

- David Hilbert,
*Mathematical problems*, Bull. Amer. Math. Soc. (N.S.)**37**(2000), no. 4, 407–436. Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. MR**1779412**, DOI https://doi.org/10.1090/S0273-0979-00-00881-8 - Harald Cramér,
*Mathematical Methods of Statistics*, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946. MR**0016588** - L. von Bertalanfy,
*The theory of open systems in physics and biology*, Science**111**(1950), 23–29. - S. Beer,
*Cybernetics and Management*, The English Universities Press, London, 1959. - V. I. Glivenko,
*Probability Theory*, Gos. uchebno-pedagog. izd-vo, Moscow, 1937. (Russian) - Karl Prachar,
*Primzahlverteilung*, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR**0087685** - Garrett Birkhoff,
*Lattice theory*, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053** - G. M. Fichtenholz,
*Differential- und Integralrechnung. I*, Twelfth edition, Hochschulbücher für Mathematik [University Books for Mathematics], vol. 61, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986 (German). Translated from the Russian by Ludwig Boll. MR**845555** - B. Z. Vulih,
*Vvedenie v teoriyu poluuporyadochennykh prostranstv*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). MR**0133668** - John C. Oxtoby,
*Measure and category*, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR**584443** - Bruno de Finetti,
*Probability, induction and statistics. The art of guessing*, John Wiley & Sons, London-New York-Sydney, 1972. Wiley Series in Probability and Mathematical Statistics. MR**0440638** - Mark Kac,
*Probability and related topics in physical sciences*, Lectures in Applied Mathematics (Proceedings of the Summer Seminar, Boulder, Colorado, vol. 1957, Interscience Publishers, London-New York, 1959. With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. MR**0102849**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
60A05

Retrieve articles in all journals with MSC (2000): 60A05

Additional Information

**Yu. I. Petunin**

Affiliation:
National Taras Shevchenko University, Faculty of Cybernetics, Volodymyrs’ka Street 64, Kyïv 01033, Ukraine

Email:
vm214@dcp.kiev.ua

**D. A. Klyushin**

Affiliation:
National Taras Shevchenko University, Faculty of Cybernetics, Volodymyrs’ka Street 64, Kyïv 01033, Ukraine

Received by editor(s):
December 17, 2001

Published electronically:
December 30, 2005

Article copyright:
© Copyright 2005
American Mathematical Society