SELECTION LEMMA FOR SEQUENCES OF MEASURABLE SETS, AND LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS

In the present paper we show that the integral functional {Mathematical expression} is lower semicontinuous with respect to the joint convergence of yk to y in measure and the weak convergence of uk to u in L1. The integrand f: G × ℝN × ℝm → ℝ, (x, z, p) → f(x, z, p) is assumed to be measurable in x for all (z,p), continuous in z for almost all x and all p, convex in p for all (x,z), and to satisfy the condition f(x,z,p)≧Φ(x) for all (x,z,p), where Φ is some L1-function. The crucial idea of our paper is contained in the following simple © 1979 Springer-Verlag.