  We discuss the classic theorem according to which a gravitational lens always
produces a total magnification greater than unity. This theorem seems to
contradict the conservation of total flux from a lensed source. The standard
solution to this paradox is based on the exact definition of the reference
'unlensed' situation.
  We calculate magnifications and amplifications for general lensing scenarios
not limited to regions close to the optical axis. In this way the formalism is
naturally extended from tangential planes for the source and lensed images to
complete spheres. We derive the lensing potential theory on the sphere and find
that the Poisson equation is modified by an additional source term that is
related to the mean density and to the Newtonian potential at the positions of
observer and source. This new term generally reduces the magnification, to
below unity far from the optical axis, and ensures conservation of the total
photon number received on a sphere around the source.
  This discussion does not affect the validity of the 'focusing theorem', in
which the unlensed situation is defined to have an unchanged affine distance
between source and observer. The focusing theorem does not contradict flux
conservation, because the mean total magnification directly corresponds to
different areas of the source sphere in the lensed and unlensed situation. We
argue that a constant affine distance does not define an astronomically
meaningful reference.
  By exchanging source and observer, we confirm that magnification and
amplification differ according to Etherington's reciprocity law, so that
surface brightness is no longer strictly conserved. [ abridged ]
