Complete Ccc Boolean Algebras

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. We also give examples of ccc forcings adding a real but not independent reals. 1. Introduction. We investigate combinatorial properties of complete ccc Bool-ean algebras. The focus is on properties related to the existence of a Maharam submeasure and on forcing properties. In particular, we address the question when the forcing adds independent reals. The work is continuation of [BGJ] and [BFH] and is related to the problems of von Neumann and Maharam. The problem of von Neumann from The Scottish Book ([Sc], Problem 163) asks whether every weakly distributive complete ccc Boolean algebra carries a countably additive measure. Von Neumann's problem can be divided into two distinctly different questions. Weak distributivity is a consequence of a property possibly weaker then measurabil-ity, namely the existence of a continuous strictly positive submeasure (a Maharam submeasure); the Control Measure Problem of [M] asks whether every complete Boolean algebra that carries a continuous submeasure must also carry a measure. For exact formulation of this question see [F]. It should be noted that the Control Measure Problem is eqiuvalent to a Π