What is in common between random or dynamically generated limit superior sets and boundaries of sets of finite perimeter. In both settings problems can be reduced to estimating the Hausdorff content of some set which has a type of self-similarity. We introduce a new lemma that yields a Hausdorff content lower bound in both of these settings, and which seems useful in further applications. We illustrate its usefulness by deriving three well known conclusions from it: the Federer characterization of sets of finite perimeter, the Hausdorff dimension of random limit superior sets and the Beresnevitch-Velani Mass Transference Principle.

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