Convergence and Extraction of Bounded Sequences inL1(R)

✍ Scribed by Heinz-Albrecht Klei

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 123 KB
• Volume: 198
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1996.0104

✦ Synopsis

We present several applications of H. P. Rosenthal's subsequence splitting 1 Ž . lemma: Each bounded sequence in L 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 1 Ž . sequence in L 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 L 1 -norm on closed convex subsets 1 Ž . of L R .