  We study the perfect Bose gas in random external potentials and show that
there is generalized Bose-Einstein condensation in the random eigenstates if
and only if the same occurs in the one-particle kinetic-energy eigenstates,
which corresponds to the generalized condensation of the free Bose gas.
Moreover, we prove that the amounts of both condensate densities are equal. Our
method is based on the derivation of an explicit formula for the occupation
measure in the one-body kinetic-energy eigenstates which describes the
repartition of particles among these non-random states. This technique can be
adapted to re-examine the properties of the perfect Bose gas in the presence of
weak (scaled) non-random potentials, for which we establish similar results.
