  We develop iterated forcing constructions dual to finite support iterations
in the sense that they add random reals instead of Cohen reals in limit steps.
In view of useful applications we focus in particular on two-dimensional
"random" iterations, which we call shattered iterations. As basic tools for
such iterations we investigate several concepts that are interesting in their
own right. Namely, we discuss correct diagrams, we introduce the amalgamated
limit of cBa's, a construction generalizing both the direct limit and the
two-step amalgamation of cBa's, we present a detailed account of cBa's carrying
finitely additive strictly positive measures, and we prove a general
preservation theorem for such cBa's in amalgamated limits. As application, we
obtain new consistency results on cardinal invariants in Cichon's diagram. For
example, we show the consistency of aleph_1 < cov (meager) < non (meager), thus
answering an old question of A. Miller.
