Simple Proofs of Some Results of Reshetnyak

In this paper we give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak. 1. Main result In 1968, Reshetnyak [20] proved two important results concerning the continuity and lower semicontinuity of functionals with respect to weak-star convergence of measures. These theorems are used in a variety of areas in the calculus of variations, ranging from problems in relaxation ([1],[3],[4],[6]) and estimates in Γ-convergence ([17],[18],[19]) to anisotropic surface energies studied in continuum mechanics ([9],[10],[11],[14]) and various other applications ([2],[7],[12]). For X a locally compact, separable metric space, let [Mb(X)] m denote the space of R-valued measures on X with finite total mass. Given μ ∈ [Mb(X)], we write |μ| for the total variation of μ and dμ d|μ| for the Radon-Nikodym derivative of μ with respect to |μ|. Under these assumptions (see Proposition 1.43 and Remark 1.57 of [5]), we have that [Mb(X)] m is the dual of [C0(X)] m (the completion of the space of R-valued continuous functions with compact support in the sup norm). Thus, for μn, μ ∈ [Mb(X)], we have that μn ∗ ⇀ μ in [Mb(X)] if