Rectifiable metric spaces: local structure and regularity of the Hausdorff measure

We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the existence of a metric differential of Lipschitzian functions also in situations where the well-known theorem of Rademacher fails. Let (X, p) be a metric space. We denote by g%n the n-dimensional Hausdorff measure over X defined by Zn(E) _ p (E) =li in f { () (diamp(Ei))n I E c U Ei, diamp(Ei) 1 , defined on subsets of Rn such that 2pn (A\Ui imagfi) = 0. It is well known (see, e.g., [4, Chapters 3.2-3.3]), that n-rectifiable subsets of Euclidian spaces exhibit many properties similar to those of n-dimensional l Secondary 26B 10.