# Stieltjes integral

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

[itex] \int_a^b f(x) \, d g(x) [itex]

without first having to convert it to

[itex] \int_a^b f(x) \, g'(x) \, dx [itex]

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

[itex] \int_a^b f(x) \, dx, [itex]

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

Thus, the form

[itex] \int_a^b f(x) \, d g(x) [itex]

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.

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