A Unifying Note on Fatou's Lemma in Several Dimensions

A general version of Fatou's lemma in several dimensions is presented. It subsumes the Fatou lemmas given by Schmeidler (Schmeidler, D. 1970. Fatou's lemma in several dimensions. Proc. Amer. Math. Soc.24 300–306.), Hildenbrand (Hildenbrand, W. 1974. Core and equilibria of a large economy. Princeton University Press, Princeton.), Cesari-Suryanarayana (Cesari, L., M. B. Suryanarayana. 1978. An existence theorem for pareto problems. Nonlinear Anal.2 225–233.), and Artstein (Artstein, Z. 1979. A note on Fatou’s lemma in several dimensions. J. Math. Econom.6 277–282.). Also, it is equivalent to an abstract variational existence result that extends and generalizes results by Aumann-Perles (Aumann, R. J., M. Perles. 1965. A variational problem arising in economics. J. Math. Anal. Appl.11 488–503.), Berliocchi-Lasry (Berliocchi, H., J.-M. Lasry. 1973. Integrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France101 129–184.), Artstein (Artstein, Z. 1974. On a variational problem. J. Math. Anal. Appl.45 404–415.), and Balder (Balder, E. J. 1979. On a useful compactification for optimal control problems. J. Math. Anal. Appl.72 391–398.) in several respects.