On the Hausdorff measure of regular ω-languages in Cantor space

This paper deals with the calculation of the Hausdorff measure of regular omega-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular omega-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which omega-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure. The first of these condition is related to the homogeneity of the local behaviour of the Hausdorff dimension of the underlying set, and the other with a certain topological density of the set in its closure.