Liftings of Pettis integrable functions
If \((\Omega,\Sigma,\mu)\) is a finite complete measure space and \(\rho\) a lifting on \(L^\infty(\mu)\), then for weakly measurable and scalarly bounded \(f:\Omega\to\) Banach space \(X\) the \(\rho_1(f): \Omega\to X^{**}\) is uniquely defined by \(x^*(\rho_1(f)(\omega))= \rho(x^*f)(\omega)\) for \(\omega\in \Omega\), \(x^*\in X^*\); \(\rho_0(f): \Omega\to X^*\) is defined similarly for \(w^*\)-measurable and \(w^*\)-bounded \(f: \Omega\to X^*\). Extending results of \textit{M. Talagrand} [``Pettis integral and measure theory'', Mem. Am. Math. Soc. 307 (1984; Zbl 0582.46049)] it is shown that if \(f\) is Pettis integrable, scalarly bounded and each \(z\in X^{***}\) is \(\mu\rho_1(f)^{-1}\)-measurable, then \(\rho_1(f)\) is Pettis integrable, too; an analogous result holds for \(\rho_0(f)\). Similarly, if \(f:\Omega\to X\) (resp. \(X^*\)) is McShane integrable [see \textit{D. H. Fremlin}, Ill. J. Math. 39, No. 1, 39-67 (1995; Zbl 0810.28006)] and scalarly bounded (\(w^*\)-bounded), \(\rho_1(f)\) (resp. \(\rho_0(f)\)) is Pettis integrable, provided \(\rho\) is consistent.
scientific article; zbMATH DE number 1539262