Convergence of measures in the random model

A Boolean algebra B has the Vitali--Hahn--Saks property if every sequence of measures on B which converges to 0 on elements of B converges also to 0 on every Borel subset of the Stone space of B. Examples of Boolean algebras having the property include e.g. sigma-complete ones. Some time ago we showed that the property is preserved for sigma-complete Boolean algebras after extending the ground model by a forcing belonging to a broad class of notions of forcing (including e.g. the Sacks, Silver or Miller forcing). In this talk, opposite to the previous result, I will show that the property is not preserved for any ground model Boolean algebra after adding a random real. This has interesting connections with cardinal characteristics of the continuum and the result of Dow and Fremlin stating that there exist Efimov spaces in the random model. (This is a joint work with L. Zdomskyy.)