facts about Riemann–Stieltjes integral

• If the integrator $g$ of the http://planetmath.org/node/3187Riemann–Stieltjes integral  ${\int }_a^{b}f(x)𝑑g(x)$ is the identity function, then the integral reduces to the Riemann integral ${\int }_a^{b}f(x)𝑑x$.

• If the integrand of the Riemann–Stieltjes integral is a constant function, one has

  $${\int }_a^{b}c𝑑g(x)=c\cdot (g(b)-g(a)).$$

• If the integrand $f$ is continuous and the integrator $g$ monotonically nondecreasing on the interval  $[a,b]$,  then there exists a number $\xi$ on the interval such that

  $${\int }_a^{b}f(x)𝑑g(x)=f(\xi )(g(b)-g(a)).$$

  Cf. the integral mean value theorem.

• If $f$ is continuous, $g$ monotonically nondecreasing and differentiable on the interval  $[a,b]$,  then