Asymptotic multiplicities and Monge-Amp\`ere masses (with an appendix by S\'ebastien Boucksom)

Ein, Lazarsfeld and Smith asked whether `equality' holds between two Samuel type asymptotic multiplicities for a graded system of zero-dimensional ideals on a smooth complex variety. We find a connection of this question to complex analysis by showing that the `equality' is equivalent to a particular case of Demailly's strong continuity property on the convergence of residual Monge-Amp\`ere masses under approximation of plurisubharmonic functions. On the other hand, in an appendix of this paper, S\'ebastien Boucksom gives an algebraic proof of the `equality' in general, using the intersection theory of b-divisors. We then use these to show that Demailly's strong continuity holds for a new important class of plurisubharmonic functions.