Stieltjes Integral

Stieltjes integral [′stēlt·yəs ‚int·ə·grəl]

(mathematics)

The Stieltjes integral of a real function ƒ(x) relative to a real functiong(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(a_i) [g(x_i) - g(x_i-1)] taken as partitions of the interval shrink. DenotedAlso known as Riemann-Stieltjes integral.

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Stieltjes Integral 

a generalization of the definite integral proposed in 1894 by T. Stieltjes. In this generalization, the limit of the Riemann sums ∑f(ξ_i)(x_i – x_i-1) is replaced by the limit of the sums ∑f(ξ_i) [Φ(x_i) – Φ(x_i-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):