Convergence theorems for the Birkhoff integral

• Autores: José Rodríguez
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 35, Nº 2, 2009, págs. 541-551
• Idioma: inglés
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• Resumen

  • We study the validity of Vitali's convergence theorem for the Birkhoff integral of functions taking values in a Banach space X. On the one hand, we show that the theorem is true whenever X has weak*-separable dual unit ball. On the other hand, we prove that if X is super-reflexive and has density character the continuum, then there is a uniformly bounded sequence of Birkhoff integrable X-valued functions (defined on [0,1] with the Lebesgue measure) converging pointwise to a non Birkhoff integrable function.