Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces

• Autores: Pedro Ortega Salvador
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 35, Nº 2, 1991, págs. 465-473
• Idioma: inglés
• DOI: 10.5565/publmat_35291_10
• Títulos paralelos:
  • Convergencia de las medias y finitud de las funciones de potencias ergódicas en espacios L1 ponderados
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• Resumen

  • Let (X, F, µ) be a finite measure space. Let T: X ? X be a measure preserving transformation and let Anf denote the average of Tkf, k = 0, ..., n. Given a real positive function v on X, we prove that {Anf} converges in the a.e. sense for every f in L1(v dµ) if and only if infi = 0 v(Tix) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Prf for every f in L1(v dµ). We apply this result to characterize, being T null-preserving, the finite measures ? for which the sequence {Anf} converges a.e. for every f Î L1(d?) and to prove that uniform boundedness of the averages in L1 is sufficient for finiteness a.e. of Pr.