Kamburelis (1989) proved that Boolean algebras with finitely additive measures preserve "strong witnesses" of the additivity of measure, which is one of the ingredients in the proof of Cichoń's maximum (forcing a constellation where all non-dependant entries in Cichoń's diagram are pairwise different).
Based on Kamburelis' results, we present a generalization of the previous preservation result by taking away the "finitely additive measure" requirement. On the other hand, we give one example where a special property on the finitely additive measure gives the preservation of the additivity of the strong measure ideal, and also develop its corresponding preservation theory that doesn't rely on finitely additive measures. This strengthens preservation results by Judah and Shelah (1989).
This is a joint work with Jörg Brendle and Miguel Cardona.