A Banach-Zarecki theorem for functions with values in Banach spaces (Q478498)

This paper gives a vector valued version of the classical Banach-Zarecki theorem, which establishes that a function \(F:[a,b] \to \mathbb R\) is strongly absolutely continuous if and only if \(F\) is of bounded variation, continuous on \([a,b]\) and satisfies the property (N), that is, for each Lebesgue measurable subset \(E\) of \([a,b]\), its Lebesgue measure \(\lambda(E)=0\) implies that \(\lambda(F(E))=0\). The vector valued version, that is proved in the paper, states the following (Theorem 2.1): Let \(X\) be a Banach space and let \(F:[0,1] \to X\) be a function with compact range. The following assertions are equivalent. {\parindent =6mm \begin{itemize}\item[(i)] \(F\) is strongly absolutely continuous with respect to the norm. \item[(ii)] \(F\) is of bounded variation, weak continuous and satisfies the weak property (N). \end{itemize}} A function \(F\) has the weak property (N) is for each \(x^* \) in the dual of \(X\), we have that the function \(x^* \circ F\), has property (N).