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Coherence of the Product Law for Independent Continuous Events

  • Daniele Mundici
Chapter
Part of the Trends in Logic book series (TREN, volume 47)

Abstract

Let \(A^*\) and \(B^*\) be finite sets of continuous events (e.g., physical observables, or random variables) represented by elements of semisimple MV-algebras A and B. Suppose \(\alpha :A^*\rightarrow [0,1]\) and \(\beta :B^*\rightarrow [0,1]\) are coherent books, i.e., maps satisfying de Finetti’s coherence criterion. Suppose all events in \(A^*\) are (logically) independent of all events in \(B^*.\) Let \(C=A\otimes B\) be the semisimple tensor product of A and B. We first prove that if \(a,a'\in A^*\) and \( b,b'\in B^*\) satisfy \(a\otimes b=a'\otimes b'\), then \(\alpha (a)\beta (b)=\alpha (a')\beta (b')\). Thus by setting \(\gamma (a \otimes b)=\alpha (a)\beta (b)\) we obtain a [0, 1]-valued function \(\gamma \) defined on the set \(C^*\) of pure tensors of C of the form \(a\otimes b\) for \(a\in A^*\) and \(b\in B^*\). We then prove that \(\gamma \) is a coherent book on \(C^*\). For the proofs we need the MV-algebraic extension of de Finetti Dutch Book theorem, Fubini theorem, and the Kroupa–Panti theorem (which in turn rests on the preservation properties of the \(\varGamma \) functor, the Stone–Weierstrass theorem and the Riesz representation theorem).

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science “Ulisse Dini”University of FlorenceFlorenceItaly

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