zbMATH — the first resource for mathematics

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.
MSC:
 28B15 Set functions, measures and integrals with values in ordered spaces