Integration by parts (Riemann–Stieltjes)

Integration by parts (Riemann–Stieltjes): Let $f,g:[a,b]\to\mathbb{R}$ be functions of bounded variation, and assume at least one of $f$ or $g$ is continuous on $[a,b]$. Then both Riemann–Stieltjes integrals $\int_a^b f\,dg$ and $\int_a^b g\,df$ exist and $\int_a^b f\,dg + \int_a^b g\,df = f(b)g(b)-f(a)g(a).$

This identity generalizes the usual integration by parts formula for Riemann integrals and is essential in applications of the Riemann–Stieltjes integral (e.g., summation by parts and Fourier analysis).

Proof sketch: For a tagged partition , the discrete sums satisfy a finite “summation by parts” identity. One shows that as the mesh of the partition goes to zero, these sums converge to the Riemann–Stieltjes integrals, and the discrete identity passes to the limit under the bounded-variation/continuity hypotheses.