Geodesic
A criterion for the approximation of a semicontinuous functional by Lipschitz functionals
Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 84-90

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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. 
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V. Ya. Prudnikov; A. G. Podgaev. A criterion for the approximation of a semicontinuous functional by Lipschitz functionals. Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 84-90. http://geodesic.mathdoc.fr/item/DVMG_2022_22_1_a7/