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Pettis integral; decomposable set; convex set; Alexiewicz norm
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.
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