Let X and Y be Banach spaces. The first part of this paper deals with a normed space which consists of weakly integrable, in a suitable sense, -valued functions. We give another norm for which is equivalent to the initial one. We provide an example when this space is not a Banach space and we prove that it is a Banach space if the measure \(\mu \) is discrete and Y is reflexive. We study some naturally defined operators on . In the second part we consider convergence theorems for sequences of functions in . Let \(\left( {{\mathscr {A}}}_t ^{(n)}\right) _{t\in \Omega }\) be a sequence in and let \(({{\mathscr {A}}}_t )_{t\in \Omega }\) be a family in such that \(\displaystyle \lim _{n\rightarrow \infty }{\mathscr {A}}_t^{(n)}={\mathscr {A}}_t\) for all \(t\in \Omega \) , where the limit is in the weak, strong or uniform sense. Under some additional conditions we prove that \(\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega {{\mathscr {A}}}_t ^{(n)}d\mu (t)=\int _\Omega {{\mathscr {A}}}_t \,d\mu (t), \end{aligned}\) where the limit is weak, strong and uniform respectively. These results generalize Dominant Convergence Theorem and Vitali Convergence Theorem. Moreover, a converse of the uniform version of Vitali Convergence Theorem is obtained.
