# An axiomatic approach to extensional probability measures

## Abstract

Replacing the demand of countable additivity (σ-additivity), imposed on probability measures by the classical Kolmogorov axiomatic, by a stronger axiom, and considering only probability measures taking their values in the Cantor subset of the unit interval of real numbers, we obtain such an axiomatic system that each probability measure satisfying these axioms is extensional in the sense that probability values ascribed to measurable unions and intersections of measurable sets are functions of probability values ascribed to particular sets in question. Moreover, each such probability measure can be set into a one-to-one correspondence with a boolean-valued probability measure taking its values in the set of all subsets of an infinite countable space, e.g., the space of all natural numbers.

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## References

- 1.A. Bundy: Incidence calculus — A mechanism for probabilistic reasoning. Journal of Automated Reasoning 1, 1985, no. 3, pp. 263–283.Google Scholar
- 2.W. Feller: An Introduction to Probability Theory and its Applications, vol. I, 2nd edition. J. Wiley and Sons, New York, 1957.Google Scholar
- 3.A. N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer-Verlag, Berlin, 1933.Google Scholar
- 4.I. Kramosil: Expert systems with non-numerical belief functions. Problems of Control and Information Theory 17 (1988), no. 5, pp. 285–295.Google Scholar
- 5.I. Kramosil: Extensional processing of probability measures. International Journal of General Systems 22 (1994), no. 2, pp. 159–170.Google Scholar
- 6.M. Loève: Probability Theory. Van Nostrand, Princeton, 1955.Google Scholar
- 7.R. Sikorski: Boolean Algebras, second edition. Springer-Verlag, Berlin-Göttingen — Heidelberg — New York, 1964.Google Scholar