Stieltjes integral

The Stieltjes integral provides a direct way of (numerically) defining an integral of the type

<math> \int_a^b f(x) \, d g(x) <math>

without first having to convert it to

<math> \int_a^b f(x) \, g'(x) \, dx <math>

and then integrating this converted form by means of a pre-existing, non-Stieltjes integration method.

Stieltjes integration provides a means of extending any type of integration of the form

<math> \int_a^b f(x) \, dx, <math>

such as Riemann integration, Darboux integration, or Lebesgue integration.

Thus, the form

<math> \int_a^b f(x) \, d g(x) <math>

can be integrated by means of Riemann-Stieltjes integration, Darboux-Stieltjes integration, or Lebesgue-Stieltjes integration. Function f is called the integrand and function g is called the integrator.

See also: Riemann-Stieltjes integral.