Hausdorff measure of infinitely generated self-similar sets

We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set A subset of or equal to R(n) with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that of A.