CCC Forcing and Splitting Reals

Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is proved using the P -ideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process, I show that under the P -ideal dichotomy every weakly distributive ccc complete Boolean algebra carries an exhaustive submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah [Sh1] it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra.