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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
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