  We give a definition of mass for conformally compactifiable initial data
sets. The asymptotic conditions are compatible with existence of gravitational
radiation, and the compactifications are allowed to be polyhomogeneous. We show
that the resulting mass is a geometric invariant, and we prove positivity
thereof in the case of a spherical conformal infinity. When R(g) - or,
equivalently, the trace of the extrinsic curvature tensor - tends to a negative
constant to order one at infinity, the definition is expressed purely in terms
of three-dimensional or two-dimensional objects.
