Riemann-Stieltjes integral (${\int }_a^{b}fdg$)

For continuous $f$ and an integrator $g$ of bounded variation, the Riemann-Stieltjes integral is the limit of tagged sums $\sum f({\xi }_j)(g(t_{j+1})-g(t_j))$ as the mesh tends to zero. It exists and equals integration against the signed measure induced by $g$. Continuous vector-valued integrators are handled componentwise, which defines path signatures of continuous bounded variation paths.