On weak convergence implying strong convergence under extremal conditions

Abstract

It is well known that under suitable extreme point conditions for the limit function, weak convergence in an L 1-space can imply strong convergence. T. Rzez̆uchowski [ Bull. Austral. Math. Soc. 39 (1989), 201–214] has shown that a similar phenomenon can occur under an extremal subset condition. We generalize his result in several respects by means of the associated Young measures, similar to the analysis given in [E. J. Balder, Bull. Austral. Math. Soc. 33 (1986), 363–368].