Coherence of the Product Law for Independent Continuous Events

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).