Vitali systems in R^n with irregular sets

Vitali type theorems are results stating that out of a given family of sets one can select pairwise disjoint sets which fill out a "large" region. Usually one works with "regular" sets such as balls. We shall establish results with sets of a more complicated geometrical structure, e.g., Cantor-like sets are allowed. The results are related to a generalisation of the classical notion of a differentiation basis.l They concern real n-space R^n and Lebesgue measure.