  Let us assume that we are given two metric spaces, where the Hausdorff
dimension of the first space is strictly smaller than the one of the second
space. Suppose further that the first space has sigma-finite measure with
respect to the Hausdorff measure of the corresponding dimension. We show for
quite general metric spaces that for any measurable surjection from the first
onto the second space, there is a set of measure zero that is mapped to a set
of positive measure (both measures are the Hausdorff measures corresponding to
the Hausdorff dimension of the first space). We also study more general
situations where the measures on the two metric spaces are not necessarily the
same and not necessarily Hausdorff measures.
