  A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric
number theory is introduced and discussed. The general conjecture is
established modulo the original conjecture. The key result is a Mass
Transference Principle which allows us to transfer Lebesgue measure theoretic
statements for lim sup subsets of R^k to Hausdorff measure theoretic
statements. In view of this, the Lebesgue theory of lim sup sets is shown to
underpin the general Hausdorff theory. This is rather surprising since the
latter theory is viewed to be a subtle refinement of the former.
