A characterization of Pettis sets in dual Banach spaces (Q1187169)

Let \(X\) be a real Banach space with topological dual \(X^*\). Let \((\Omega,\Sigma,\mu)\) be a complete probability measure space and \(([0,1],\Lambda,\lambda)\) the Lebesgue measure space on \([0,1]\). If \(f: \Omega\to X^*\) is a weak\(^*\)-measurable function with bounded range, then the bounded linear operator \(T_ f: X\to L_ 1(\Omega,\Sigma,\mu)\) is defined by \(T_ f(x)=x\circ f\) for \(x\in X\). The dual operator of \(T_ f\) is denoted by \(T_ f^*\). Then the following theorem is established. Theorem. Let \(H\) be a weak\(^*\)-compact subset of \(X^*\). Then the following statements are equivalent. (a) The set \(H\) is a Pettis set. (b) For any weak\(^*\)-measurable function \(f:[0,1]\to H\), \(\{T_ f^*(\chi_ A/\lambda(A))\): \(\lambda(A)>0\), \(A\in\Lambda\}\) contains no \(\delta\)-Rademacher tree. (c) For any \((\Omega, \Sigma, \mu)\) and any weak\(^*\)-measurable function \(f: \Omega \to H\), \(\{T_ f^*(\chi_ E/\mu(E))\): \(\mu(E)>0\), \(E\in \Sigma\}\) contains no \(\delta\)-Rademacher tree.