  The proof of the Luttinger theorem, which was originally given for a normal
Fermi liquid with equal spin populations formally described by the exact
many-body theory at zero temperature, is here extended to an approximate theory
given in terms of a "conserving" approximation also with spin imbalanced
populations. The need for this extended proof, whose underlying assumptions are
here spelled out in detail, stems from the recent interest in superfluid
trapped Fermi atoms with attractive inter-particle interaction, for which the
difference between two spin populations can be made large enough that
superfluidity is destroyed and the system remains normal even at zero
temperature. In this context, we will demonstrate the validity of the Luttinger
theorem separately for the two spin populations for any "$\Phi$-derivable"
approximation, and illustrate it in particular for the self-consistent
$t$-matrix approximation.
