An integral representation of functions in gas-kinetic models

Motivated by the theory of kinetic models in gas dynamics, we obtain an integral representation of lower semi-continuous functions on R-d, d >= 1. We use the representation to study the problem of compactness of a family of the solutions of the discrete time BGK model for the compressible Euler equations. We determine sufficient conditions for strong compactness of moments of kinetic densities, in terms of the measures from their integral representations.