The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of
the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the waves functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice of coordinates. The constructions are illustrated in the
examples of ultrastatic space-times and de Sitter space-time.