Stieltjes Integral

Stieltjes integral

[′stēlt·yəs ‚int·ə·grəl] (mathematics) The Stieltjes integral of a real function ƒ(x) relative to a real function g (x) of bounded variation on an interval [a,b ] is defined, analogously to the Riemann integral, as a limit of a sum of terms ƒ(ai ) [g (xi ) - g (xi-1)] taken as partitions of the interval shrink. Denoted Also known as Riemann-Stieltjes integral.

Stieltjes Integral

 

a generalization of the definite integral proposed in 1894 by T. Stieltjes. In this generalization, the limit of the Riemann sums ∑f(ξi)(xi – xi-1) is replaced by the limit of the sums ∑f(ξi) [Φ(xi) – Φ(xi-1)]. where the integrating function ϕ(x) is a function of bounded variation (seeVARIATION OF A FUNCTION). If ϕ(x) is differentiable, then the Stieltjes integral can be expressed in terms of the Riemann integral (if it exists):