Compactness in the space ${L^{p}}$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin, J.L. Lions, J. Simon, and, more recently, by J.M. Rakotoson and R. Temam, who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of “tightness” and “integral equicontinuity”, new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin - Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved
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