Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann

It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure

COMPLETE CCC BOOLEAN ALGEBRAS, THE ORDER SEQUENTIAL TOPOLOGY, AND A PROBLEM Of Von Neumann

Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. 1. Either there exists a Maharam submeasure on B or every nonempty open set in (B,τs) is topologically dense. 2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure. 3. The topological space (B,τs) is sequentially compact if and only if the generic extension by B does not add independent reals. Examples are also given of ccc forcings adding a real but not independent reals