A theorem on lower semicontinuity of integral functionals

A general lower semicontinuity theorem, in which not only mappings uM and PM but also the integrands fM depend on M, is proved for integrands f, fM under certain general hypotheses including that f(x, u, P) is convex respect to P and fM converge to f locally uniformly, but fM (x, u, P) are not required to be convex respect to P and fM (x, ·, ·) do not even need to be lower semicontinuous. Some more usable criteria, as corollaries of the main theorem, for lower semicontinuity of integral functionals are also given