Stieltjes Integral

The Stieltjes integral provides a direct way of (numerically) defining an integral of the type
\int_a^b f(x) \, d g(x)
without first having to convert it to
\int_a^b f(x) \, g'(x) \, dx
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
\int_a^b f(x) \, dx,
such as Riemann integration, Darboux integration, or Lebesgue integration. Thus, the form
\int_a^b f(x) \, d g(x)
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.

 

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