On the \(L^p\) space of observables on product MV algebras (Q1586481)

Acording to [\textit{B. Riečan}, ``On the product MV algebras'', Tatra Mt. Math. Publ. 16, 143-149 (1999; Zbl 0951.06013)], a product MV algebra is an MV-algebra \((M,\oplus,\odot,\ast,0,1)\) considered together with an operation \(\cdot\) that turns \(M\) into a commutative semigroup satisfying some additional conditions formulated in terms of the commutative \(l\)-group corresponding to \(M\). Observables on \(M\) are defined in the usual fashion as certain mappings from the \(\sigma\)-algebra of Borel subsets of \(R\) into \(M\). An observable \(x\) is said to belong \(L^p\) if its norm \(\int_R |t|dm_x(t)\) exists. The author proves in the paper that, for \(M\) weakly distributive, \(L^p\) becomes a complete pseudometric space under an appropriate (and naturally defined) distance function.