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A finitisation of the finitely additive probability theory using non standard analysis. (English) Zbl 0831.60004
Summary: We prove, within the frame of nonstandard analysis, that the finitely additive probability theory of B. de Finetti is equivalent to the elementary probability theory on finite spaces. This equivalence reduces the solution of various classical problems to purely combinatorial constructions. We use it to get a new insight into comparison of zero- probabilities, the extension of conditional probability laws, equiprobability on infinite sets etc…In particular, we get a finitely additive probability law on the power set of the Euclidean real line, which is invariant under all isometries. In order to get the paper self-contained, we give in the appendix a brief account on E. Nelson’s axiomatic setting of nonstandard analysis called internal set theory [Bull. Am. Math. Soc. 83, 1165-1198 (1977; Zbl 0373.02040)].

60A99 Foundations of probability theory
03H05 Nonstandard models in mathematics