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

Preservation properties for iterations with finite support