Integration by parts for the Denjoy-Bochner integral (Q1434315)

Let \(X\) be a Banach space and \(f: [a, b]\to X\). Then \(f\) is said to be Denjoy-Bochner integrable on \([a, b]\) if there exists an ACG function \(F: [a, b]\to X\) such that \(F\) is approximately differentiable a.e. on \([a, b]\) and \(F_{ap}'= f\) a.e. on \([a, b]\). In this paper, integration by parts formulae and a Riesz-type representation theorem are proved for Denjoy-Bochner integrals.