= Riemann-Stieltjes integral
{c}
{title2=$\int_a^b f\,dg$}
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, of
$$
\sum_i f(\xi_i)\bigl(g(x_i)-g(x_{i-1})\bigr),\qquad \xi_i\in[x_{i-1},x_i].
$$
Unlike 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>.
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