Measure-free conditioning and extensions of additive measures on finite MV-algebras
As was shown by several chapters of the second volume of [\textit{E. Pap} (ed.), Handbook of measure theory. Amsterdam: North-Holland (2002; Zbl 0998.28001)], MV-algebraic measure theory has reached a mature stage. Most of Carathéodory's Boolean algebraic probability theory can be extended to MV-algebras. Generalizing a celebrated result of de Finetti, the present reviewer has shown that MV-algebraic states generalize to continuous events the notion of coherent probability assessment. Further, the Kroupa-Panti theorem shows that the (finitely additive) states on any MV-algebra \(A\) are in one-one correspondence with the regular (countably additive) Borel measures on the maximal spectral space of \(A\). For recent developments on MV-algebraic measure theory, see the present reviewer's book [Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic -- Studia Logica Library 35. Berlin: Springer (2011; Zbl 1235.03002)]. The paper under review deals with algebras of conditional events and their measures. Given a Boolean algebra \(B\) of events, the conditional event ``\(a\) given \(b\)'' can be defined as the interval \(|a\,\&\,b,b\to a|\) in \(B\). The set of conditional events then turns out to be naturally equipped with the structure of an MV-algebra \(B'\). Any state of \(B\) can be extended to an operation on \(B'\) having some sort of additivity property. More generally, when the set of events forms an MV-algebra \(M\), one can similarly construct an enveloping algebra \(M'\) of conditionals events having the structure of a commutative residuated lattice-ordered involutive monoid, and called ``Girard algebra'' in the present paper. For \(M\) a finite MV-algebra, the author investigates the extendability of each state of \(M\) to an operation on \(M'\) satisfying suitable additivity properties.
scientific article; zbMATH DE number 5843027
scientific article