A long standing question in cosmology is whether gravitational lensing changes the distance-redshift relation D ( z ) or the mean flux density of sources .
Interest in this has been rekindled by recent studies in non-linear relativistic perturbation theory that find biases in both the area of a surface of constant redshift and in the mean distance to this surface , with a fractional bias in both cases on the order of the mean squared convergence \langle \kappa ^ { 2 } \rangle .
Any such area bias could alter CMB cosmology , and the corresponding bias in mean flux density could affect supernova cosmology .
Here we show that , in an ensemble averaged sense , the perturbation to the area of a surface of constant redshift is in reality much smaller , being on the order of the cumulative bending angle squared , or roughly a part-in-a-million effect .
This validates the arguments of Weinberg ( 1976 ) that the mean magnification \mu of sources is unity and of Kibble & Lieu ( 2005 ) that the mean direction-averaged inverse magnification is unity .
It also validates the conventional treatment of lensing in analysis of CMB anisotropies .
But the existence of a scatter in magnification will cause any non-linear function of these conserved quantities to be statistically biased .
The distance D , for example , is proportional to \mu ^ { -1 / 2 } so lensing will bias \langle D \rangle even if \langle \mu \rangle = 1 .
The fractional bias in such quantities is generally of order \langle \kappa ^ { 2 } \rangle , which is orders of magnitude larger than the area perturbation .
Claims for large bias in area or flux density of sources appear to have resulted from misinterpretation of such effects : they do not represent a new non-Newtonian effect , nor do they invalidate standard cosmological analyses .