# A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals

- Received:

DOI: 10.1007/BF01297738

- Cite this article as:
- Eisen, G. Manuscripta Math (1979) 27: 73. doi:10.1007/BF01297738

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## Abstract

In the present paper we show that the integral functional\(I(y,u): = \int\limits_G {f(x,y(x),u(x))dx} \) is lower semicontinuous with respect to the joint convergence of y_{k} to y in measure and the weak convergence of u_{k} to u in L_{1}. The integrand f: G × ℝ^{N} × ℝ^{m} → 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 L_{1}-function.

The crucial idea of our paper is contained in the following simple