 Riemann-Stieltjes integral

For a continuous function $f:[a,b]\to \mathbb{R}$ and an integrator $g$ of bounded variation, the Riemann-Stieltjes integral is the common limit, as the mesh of a partition tends to zero, ofUnlike an ordinary Riemann sum, it weights each sample by the change in $g$, rather than by the interval length. Uniform continuity of $f$ and the finite total variation of a function bound make the tagged sums Cauchy: passing to a common partition refinement changes a sum by at most the modulus of continuity of $f$ at its mesh times the total variation of a function of $g$.

If $g(x)=x$, this is the ordinary Riemann integral. If $g$ has a single unit jump at an interior point $c$ and is otherwise constant, the value is $f(c)$: all its weight concentrates at that point. For a continuous nondecreasing $g$, the associated Lebesgue-Stieltjes measure, with no atom at $a$, satisfies $\mu ((u,v])=g(v)-g(u)$ and gives the same integral. More general discontinuous integrators require attention to endpoint and one-sided conventions when identifying their signed measures.

Vector-valued integrators are treated componentwise. In particular, a continuous path of bounded variation permits the iterated Riemann-Stieltjes integrals defining its path signature.