  We present the classical theory of preservation of $\sqsubset$-unbounded
families in generic extensions by ccc posets, where $\sqsubset$ is a definable
relation of certain type on spaces of real numbers, typically associated with
some classical cardinal invariant. We also prove that, under some conditions,
these preservation properties can be preserved in direct limits of an
iteration, so applications are extended beyond the context of finite support
iterations. Also, we make a breve exposition of Shelah's theory of forcing with
an ultrapower of a poset by a measurable cardinal.
