The purpose of the research is to study the Bergman function and Bergman distance to generalize Moreau – Yosida Approximation. To do that we replace the quadratic additive terms in Moreau – Yosida Approximates by more general Bergman distance and study properties of generalized approximation and prove equivalence between epigraph – convergence and pointwise convergence of the generalized Moreau – Yosida Approximation.