Nets and sequences of \(p\)-Bochner, \(p\)-Dunford and \(p\)-Pettis integrable functions with values in a Banach space
Let \((\Omega,\Sigma,\mu)\) be a finite measure space and \(X\) a real Banach space. For \(p\in[1,\infty)\), a function \(f:\Omega\rightarrow X\) is said to be \(p\)-Bochner (resp. \(p\)-Dunford) integrable if it is measurable (resp. scalarly measurable) and \(\Vert f\Vert\in L_p(\mu)\) (resp. \(x^*f\in L_p(\mu)\) for each \(x^*\in X^*\)); it is said to be \(p\)-Pettis integrable if it is \(p\)-Dunford integrable and Pettis integrable. A net \(\{f_\alpha\}_{\alpha\in I}\) of \(p\)-Bochner (resp. \(p\)-Dunford) integrable functions is \(\delta\)-Cauchy if for each \(\varepsilon > 0\), there exist \(\gamma\in I\) and \(\delta > 0\) such that \(\int_E\Vert f_\alpha-f_\beta\Vert^p \,d\mu <\varepsilon\) (resp. \(\sup\{ \int_E|x^*(f_\alpha-f_\beta)|^p \,d\mu: x^*\in X^*,\,\Vert x^*\Vert\leq 1\} <\varepsilon\)) for all \(\alpha,\beta\in I\) with \(\alpha,\beta\geq \gamma\) and for all \(E\in\Sigma\) with \(\mu(E) < \delta\). The authors study for nets in detail the relationship between different types of convergence, in particular between \(\delta\)-Cauchy convergence, (Cauchy) convergence in measure, scalar equi-convergence or equi-Cauchy convergence in measure, (Cauchy) convergence in \(p\)-Bochner or \(p\)-Pettis norm, and uniform integrability. Moreover, a criterion is given for a net \(\{f_\alpha\}_{\alpha\in I}\) of \(p\)-Pettis integrable functions scalarly convergent in measure to \(f\) so that \(f\) is \(p\)-Pettis integrable and \(\{\text{Pettis-}\int_Ef_\alpha \,d\mu\}_{\alpha\in I}\) converges weakly to \(\text{Pettis-}\int_Ef\,d\mu\) for \(E\in\Sigma\).
scientific article; zbMATH DE number 7344686
scientific article