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Probability on MV-algebras. (English) Zbl 1017.28002
Pap, E. (ed.), Handbook of measure theory. Vol. I and II. Amsterdam: North-Holland. 869-909 (2002).
This handbook chapter brings a survey on MV-algebraic generalizations of the classical probability theory and it is splitted into four sections: 1) Background on MV-algebras; 2) States and observables; 3) MV-algebras with product; 4) Finitely additive measures. The first section provides a necessary algebraic background, including basic MV-algebras notions, several representation theorems (e.g., of semisimple MV-algebras due to L. P. Belluce [Can. J. Math. 38, 1356-1379 (1986; Zbl 0625.03009)]), Loomis-Sikorski theorem for MV-algebras, etc. The second part deals with states and observables in MV-algebras, discusses the independence and MV-algebras variants of some famous probabilistic theorems (central limit theorem, laws of large numbers, etc.). The special case of MV-algebras with product is discussed in the third section, including the conditional expectation, ergodic theorem, etc. The last section deals with the entropy of dynamical systems and the entropy of full tribes. Each section is endowed with bibliographical remarks containing a lot of additional information. Finally, 7 open problems are included. Note that the topic of this survey chapter is still a “hot” one and thus this chapter is not only a nice recapitulation of the state-of-art, but first of all a challenge for interested researchers to complete the MV-algebraic probability theory.
For the entire collection see [Zbl 0998.28001].

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
26E50 Fuzzy real analysis
06D35 MV-algebras
37B40 Topological entropy