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A boundary-term identity relating two Riemann–Stieltjes integrals.
Integration by parts (Riemann–Stieltjes): Let . Assume are of bounded variation and that at least one of or is continuous. Then both Riemann–Stieltjes integrals and exist, and
This generalizes the usual integration by parts (obtained by taking and interpreting ), and existence is ensured by the Riemann–Stieltjes integrability theorem.
A function of bounded variation on is a function whose total variation on is finite, i.e.
where the supremum is taken over all partitions of .
Functions of bounded variation are important because they provide a broad class of integrator functions for the Riemann–Stieltjes integral and admit structural decompositions such as the Jordan decomposition.
Integration by parts: Let and let be functions that are differentiable on , with and Riemann integrable on . Then
Riemann–Stieltjes integrability theorem: Let . If is continuous and is a bounded variation function, then is Riemann–Stieltjes integrable with respect to the integrator function on .
Since every monotone function has bounded variation, this theorem extends existence of the Riemann integral (the case ) to a wider class of integrators.