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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