Copies of \(\ell _{\infty }\) in the space of Pettis integrable functions with integrals of finite variation (Q2895971)

Let \((\varOmega ,\varSigma ,\mu)\) be a complete finite measure space and \(X\) be a Banach space. The space of all classes of scalarly equivalent weakly \(\mu \)-measurable \(X\)-valued Pettis integrable functions is denoted by \(\mathcal P_1(\mu, X)\). For every \(f \in \mathcal P_1(\mu, X)\) denote by \(Sf\) the following \(X\)-valued measure: \((Sf)(A) = \int_A f d \mu\). Finally, \(\mathcal M (\varSigma,\mu, X)\) stands for the space of those \(f \in \mathcal P_1(\mu, X)\) whose \(Sf\) has bounded variation. The author proves that \(\mathcal M (\varSigma ,\mu, X)\) equipped with the variation of \(Sf\) on \(\varOmega\) as the norm (Musiał space) contains an isomorphic copy of \(\ell_{\infty }\) if and only if \(X\) does. The result continues the research of the author from [\textit{J. C. Ferrando}, J. Math. Anal. Appl. 380, No. 1, 323--326 (2011; Zbl 1228.46014)].