A note on strong convergence for Pettis integrable function,

In a recent paper Amrani, Castaing and Valadier obtained a convergence result for a sequence of Pettis integrable functions that continues a series of results started by Olech, Tartar and Visintin. These results are of the following type. Given a weakly convergent sequence of integrable functions, one formulates an extreme point condition (pointwise) for the limit function. This eliminates persistent oscillations, and forces the convergence to the limit function to be strong as well (for instance, convergence in measure). By the use of Young measures, Visintin’s results were improved in and and extended to Bochner integrable functions with values in a separable Banach space. In this note it is shown that Theorem 2.4, the main result of Amrani-Castaing-Valadier, can be approached in a manner that is very similar to the method introduced in a paper of Balder. This leads to a considerable improvement of that result, and in a direction that is suggested on p. 331 of Amrani-Castaing-Valadier.