# Coherence of the Product Law for Independent Continuous Events

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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