## Generated envelopes.(English)Zbl 0777.60004

Denote by B a Boolean algebra and by $$P$$ the set of finitely additive probabilities in B. We call an initial valuation any map $$v$$ from B in $$[0,1]$$; the meaning of $$v$$ is that the available information about a random phenomenon is of type “the actual probability of the event $$e$$ is at least $$v(e)$$”. Following A. Papamarcou and T. L. Fine [Ann. Probab. 14, 710-723 (1986; Zbl 0595.60003)], we call $$v$$ dominated if a probability exists greater than or equal to $$v$$. If this is the case, we consider the valuation $$\overline{v}$$ defined by $\overline{v}(e)=\text{Inf}\{p(e) |\;p\in P,\;p\geq v\}, \qquad e\in\mathbf{B}.$ This new valuation, we call lower envelope generated by $$v$$, is an improvement of the initial valuation, but the formula defining $$\overline{v}$$ is not constructive since it refers to the whole class of probabilities greater than $$v$$. We furnish a formula to obtain $$\overline{v}$$ from below, that is from the values taken by $$v$$. The same question was examined by K. Weichselberger and S. Pöhlmann [A methodology for uncertainty in knowledge-based systems (1990; Zbl 0705.68097)] in particular cases. Also a necessary and sufficient condition for $$v$$ to be dominated is established and a characterization of the lower envelopes is obtained.
Reviewer: L.Biacino (Napoli)

### MSC:

 60A99 Foundations of probability theory

### Citations:

Zbl 0595.60003; Zbl 0705.68097
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