Vitali type convergence theorems for Banach space valued integrals (Q381077)

The main body of the paper consists of the Sections 2 and 3. The following notations are used: \((\Omega, \Sigma, \mu)\) stands for a complete probability space and \(X\) for a Banach space. Let \(f_n\), \(n \in\mathbb N\), and \(f\) be \(X\)-valued scalarly measurable functions on \(\Omega\). In Section 2, the authors introduce what they call the scalar equi-convergence in measure of \((f_n)\) to \(f\). It is implied by convergence in measure provided that the functions under consideration are strongly measurable. The converse fails if \(X\) is infinite-dimensional, \({\Omega} = [0, 1]\), \({\Sigma}\) is the Lebesgue \(\sigma\)-algebra and \(\mu\) is the Lebesgue measure. This is seen by analysing Example 1 of \textit{S. J. Dilworth} and \textit{M. Girardi} [Contemp. Math. 144, 69--80 (1993; Zbl 0802.46052)]. Suppose now that the \(f_n\) are Pettis integrable. Using the notion introduced, the authors establish a necessary and sufficient condition that \(f\) is also Pettis integrable and \((f_n)\) converges to \(f\) in the Pettis norm (Theorem 2.4). This is a version of a related result due to \textit{J. Rodríguez} [Proc. Am. Math. Soc. 137, No. 1, 235--245 (2009; Zbl 1185.28018), Theorem 2.8]. In Section 3, convergence with respect to an ideal over \(\mathbb N\) of sequences of both scalar- and \(X\)-valued functions on \(\Omega\) is discussed and Lebesgue and Vitali type theorems are established. In particular, a part of Theorem 2.4 is generalized. Reviewer's remark: It seems that no use is made in the paper of the completeness assumption on the measure space \((\Omega, \Sigma, \mu)\).