A structural approach to solving the 6th Hilbert problem. (Ukrainian, English) Zbl 1097.60001

Petunin, Yu. I.; Klyushin, D. A.

Teor. Jmovirn. Mat. Stat. 71, 145-159 (2004); translation in Theory Probab. Math. Stat. 71, 165-179 (2005).

The authors consider the field of random events as a partially ordered set endowed with a natural order of random events. The frequency definition of probability is formalized and modified. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The distribution of the field of events generated by random variables is studied. It is proved that, in the case of continuous random variables, the probability distribution generated by random variables is not a measure, but only finitely additive function of events.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60A05

Axioms; other general questions in probability

Keywords:

frequency approach; definition of probability; complete totally distributive Boolean algebra