Convergence and Extraction of Bounded Sequences in L1 R.

We present several applications of H. P. Rosenthal’s subsequence splitting lemma: Each bounded sequence in L1 R. admits a subsequence satisfying the conclusion of a generalized Fatou’s lemma and presenting a concentration of mass phenomenon. We show that the modulus of uniform integrability of a bounded sequence in L1 R. plays a capital role in the convergence in measure of such a sequence. A Cauchy type theorem for the convergence in measure is established. Finally we study the existence of minima of the L1-norm on closed convex subsets of L1 Rq..