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Integration by parts in vector lattices. (English) Zbl 0982.28501
Let \(X\) and \(Y\) be vector lattices and \(L_0(X, Y)\) the space of order-continuous linear operators from \(X\) into \(Y\). Let \(T= [a,b]\), \(f: T\to X\) and \(g:T\to L_0(X,Y)\). Using order convergence the authors define in a natural way the Riemann-Stieltjes integrals \(\int_T f dg\) and \(\int_T g df\) and establish an integration by parts formula for these integrals analogous to the classical scalar formula. The authors then show that the integral \(\int_Tf dg\) exists when \(f\) is uniformly order continuous, \(g\) is of order bounded variation and \(Y\) is \(\sigma\)-complete. An application to Fourier series is also given.
28B15 Set functions, measures and integrals with values in ordered spaces