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Keywords:
Pettis integral; decomposable set; convex set; Alexiewicz norm
Pettis integral; decomposable set; convex set; Alexiewicz norm
Summary:
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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