  The recently suggested notion of total mass density for closed universes is
extended to closed universes with a positive cosmological constant. Assuming
that the matter fields satisfy the dominant energy condition, it is shown that
the cosmological constant provides a sharp lower bound for the total mass
density, and that the total mass density takes this as its minimum value if and
only if the spacetime is locally isometric with the de Sitter spacetime. This
notion of total mass density is extensible to non-compact three-spaces of
homogeneity of Bianchi class A cosmological spacetimes.
