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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.

##### MSC:
 60A10 Probabilistic measure theory 06D35 MV-algebras 28E10 Fuzzy measure theory
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##### References:
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