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Conditional probability on MV-algebras. (English) Zbl 1061.60004
D. Mundici and B. Riečan [in: Handbook of measure theory. Vol. I and II, 869–909 (2002; Zbl 1017.28002)] asked whether it is possible to generalize conditional probability to MV-algebras with product (additional operation). In the paper under review a positive answer is given. The solution heavily depends on the so-called Loomis-Sikorski theorem for MV-algebras [proved by D. Mundici, Adv. Appl. Math. 22, No. 2, 227–248 (1999; Zbl 0926.06004) and by A. Dvurečenskij, J. Aust. Math. Soc., Ser. A 68, No 2, 261–277 (2000; Zbl 0958.06006)] which allows to translate the construction of a conditional state to Łukasiewicz tribes.
Let \(M\) be a \(\sigma\)-complete MV-algebra with product, let \(m\) be a state on \(M\), and let \(N\) be a \(\sigma\)-complete sub-MV-algebra of \(M\). A conditional state \(m(a\mid N)\), \(a\in M\), is a measurable function in the tribe corresponding to \(M\) such that the integral of \(m(a\mid N)\) is equal to \(m(a\cdot b)\) for all \(b\in N\) (here \(a\cdot b\) is the product of \(a\) and \(b\) in \(M\)). It is proved that \(m(a\mid N)\) has nice properties, it generalizes the conditional probability in a natural way, and if \(M\) is a tribe, then the construction of \(m(a\mid N)\) is much more simple. Some notes about conditioning for fuzzy sets are included. A trivial misprint occurs in Proposition 6.

60A10 Probabilistic measure theory
06D35 MV-algebras
28E10 Fuzzy measure theory
Full Text: DOI
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