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On the conditional expectation on probability MV-algebras with product. (English) Zbl 1003.60010

Let \(M\) be a \(\sigma\)-complete MV-algebra with product, let \(m\) be a faithful state on \(M\), and let \((M,m)\) be the resulting probability MV-algebra. An observable is a map of the Borel subsets of \(R\) into \(M\) partially preserving the structure of events. Let \(x\) and \(y\) be observables. The conditional expectation \(E(x|y)\) is constructed and the following variant of the martingale convergence theorem is proved: Let \((g_n)\) be a sequence of Borel measurable functions, let \(y_n = y\circ g^{-1}_{n}\), \(n\in N\). Then, under natural additional conditions, \(E(x|y_n)\) converges to \(E(x|y)\) almost everywhere with respect to the distribution \(m_y\). The constructions generalize the corresponding results for MV-algebras of fuzzy sets [B. Riecan, Soft Comput. 4, No. 1, 49-57 (2000)].

MSC:

60B99 Probability theory on algebraic and topological structures
28A25 Integration with respect to measures and other set functions
06D35 MV-algebras
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