On the Boolean algebra tensor product via Caratheodory spaces of place functions

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin’s Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin’s problem 315Y(f) in \cite{fremlin_measure} concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions.