  For arbitrary Borel probability measures on the real line, necessary and
sufficient conditions are presented that characterize best purely atomic
approximations relative to the classical Levy probability metric, given any
number of atoms, and allowing for additional constraints regarding locations or
weights of atoms. The precise asymptotics (as the number of atoms goes to
infinity) of the approximation error is identified for the important special
cases of best uniform (i.e., all atoms having equal weight) and best
(unconstrained) approximations, respectively. When compared to similar results
known for other probability metrics, the results for Levy approximations are
more complete and require fewer assumptions.
