Abstract:In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left | \left \|\cdot \right \| \right |$ \big (where $\left | \left \| f\right \| \right | =\sup _{[a,b] \subset [0,1]} \big \| \int _{a}^{b}f(s) ds \big \|$\big ) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.
Keywords: Pettis integral, decomposable set, convex set, Alexiewicz norm
AMS Subject Classification: 46A20, 46E30, 52A07, 54A10