Resumen de Convergence theorems for the Birkhoff integral

José Rodríguez

• 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.