  We demonstrate that the factorization of the tunneling transmission into the
product of two surface transmission functions and a vacuum decay factor allows
one to generalize Julliere's formula and explain the meaning of the ``tunneling
density of states'' in some limiting cases. Using this factorization we
calculate spin-dependent tunneling from clean and oxidized fcc Co surfaces
through vacuum into Al using the principal-layer Green's function approach. We
demonstrate that a monolayer of oxygen on the Co (111) surface creates a
spin-filter effect due to the Co-O bonding which produces an additional
tunneling barrier in the minority-spin channel. This changes the minority-spin
dominated conductance for the clean Co surface into a majority spin dominated
conductance for the oxidized Co surface.
