Influence of gravitational lensing on the determination of the   luminosity function of background objects
  The distribution function of quasars with respect to apparent brightness is given, found under the assumption that quasars are, at least partially, the gravitationally enhanced images of the active nuclei of distant galaxies. A Schechter law and a two-power law for the luminosity function of the sources are used, as well as a probabilistic law of image enhancement for various models of gravitational lenses. To find the theoretical distribution function of quasars with respect to apparent brightness we use a theorem on the probability density of a product of random quantities. It is shown that the slope of this function ranges from -1 to -2 for faint quasars, like that for ordinary galaxies. In the case of bright quasars, the slope of the apparent brightness distribution function is determined mainly by the lensing effect and has a lower limit of -3. The good agreement between theory and observations suggests that statistically quasars are gravitationally enhanced images of the active nuclei of distant galaxies. If the initial assumptions are correct, then the luminosity function of galaxies and the apparent brightness function for quasars are not independent but are related by means of the differential lensing probability. 
Yu. L. Bukhmastova