Let \((\Omega,\Sigma,\mu)\) be a complete finite measure space and \(X\) be a Banach space isomorphic to a closed linear subspace of \(l_{\infty}.\) Let \(f_n:\Omega\rightarrow X\) be a sequence of Birkhoff integrable functions converging pointwise in norm to a function \(f:\Omega\rightarrow X\). In this paper, the author shows (in Corollary 2.9) that \(\{f_n:n\in\mathbb{N}\}\) is equi-Birkhoff integrable if and only if \(f\) is Birkhoff integrable and for each \(A\in\Sigma\), \(\int_Af_nd\mu\rightarrow\int_Afd\mu\) in norm. In order to prove this result, he also proves an interesting “Vitali-type” convergence result (in Theorem 2.8) for the Pettis integral. He has also shown by a counter-example (Example 2.10) that the assumption “\(X\) is isomorphic to a subspace of \(l_\infty\)” made in Corollary 2.9 cannot be dropped. But for arbitrary Banach spaces, the author has proved the following interesting result (in Theorem 2.12) with respect to the weak topology. Let \(f_n:\Omega\rightarrow X\) be a sequence of functions converging pointwise in the weak topology to a function \(f:\Omega\rightarrow X\). If \(\{f_n:n\in\mathbb{N}\}\) is equi-Birkhoff integrable, then \(f\) is Birkhoff integrable and for each \(A\in\Sigma\), \(\int_Af_nd\mu\rightarrow\int_Afd\mu\) weakly.
This paper definitely makes a significant contribution to the modern theory of vector integration, in particular to the theory of the Birkhoff integral.