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

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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).

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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