On certain decomposition of bounded weak\(^*\)-measurable functions taking their ranges in dual Banach spaces (Q1378304)

Let \(X\) be a real Banach space and let \(X^*\) and \(X^{**}\) be its dual and bidual, respectively. Let \((S,\Sigma,\mu)\) be a complete finite measure space. Let \(C\) be a subset of \(X^{**}\). A Gelfand measurable function \(f: S\to X^*\) is said to be \(C\)-Pettis integrable if for each \(x^{**}\in C\), \(x^{**}\circ f\in L_1(S,\Sigma, \mu)\) and for each \(E\in\Sigma\), \((x^{**}, T^*_f(\chi_E))= \int_E (x^{**},f(s))d\mu(s)\) for each \(x^{**}\in C\), where \(T_f: X\to L_1(S, \Sigma,\mu)\) is given by \(T_fx= x\circ f\) and \(T^*_f\) is the dual operator of \(T_f\). A weak\(^*\) measurable function \(f: S\to X^*\) is said to be \(C\)-Pettis decomposable if there exist functions \(g\) and \(h\) such that \(g\) is \(C\)-Pettis integrable and \(h\) is weak\(^*\) scalarly null (in the sense that \((x,f(s))= 0\) \(\mu\)-a.e. on \(S\) for each \(x\in X\)) and such that \(f= g+ h\). The present paper gives a geometric condition in terms of weak\(^*\) dentability to ensure that a bounded weak\(^*\) measurable function \(f: S\to X^*\) be \(C\)-Pettis decomposable. Some related results are deduced from the above result.