A criterion for the approximation of a semicontinuous functional by Lipschitz functionals

It is proved in [1, 2] that a functional semi-continuous from below and bounded from below in the metric space X is represented as the limit of a non-decreasing family of Lipschitz functionals. In the lemma from [3], a sufficient condition for such a representation is given for a function semi-continuous from below with respect to one of the variables in a finite-dimensional space. This paper contains a criterion for approximation of a semi-continuous functional from below in a metric space by Lipschitz functionals.