Stieltjes integral
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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.