  We show that the Carath\'{e}odory 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 Carath\'{e}odory 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 Carath\'{e}odory spaces of place functions.
