In measure-theoretic analysis and related branches of mathematics, the **Lebesgue-Stieltjes integration** generalizes the Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

Lebesgue-Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue-Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory of the present topic is due. They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory.

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In order to define the Lebesgue-Stieltjes integral, we will begin by associating a measure, μ_{w}, with a non-negative, additive function of an interval, *w*(*I*), which is of bounded variation. Let (Ω, ** F**) be a measurable space such that

We may now proceed to construct the Lebesgue-Stieltjes integral of a non-negative, measurable function in a similar fashion to the construction of the corresponding Lebesgue integral. If (Ω, ** F**, μ

It is often required, of course, to compute the integral of arbitrary measurable functions *f*:(Ω, ** F**) →

We are finally equipped to define the Lebesgue-Stieltjes integral of an arbitrary function *f* with respect to the measure associated with an arbitrary additive function of an interval, *v*, which is of bounded variation.

Letg= max(0,f) andh= max(-f, 0), and letw_{1}andw_{2}be the upper and lower variations ofv, respectively. Then if μ_{v}is defined according to equations (1) and (3), theLebesgue-Stieltjes integraloffwith respect to μ_{v}iswhere each of the integrals on the right hand side of this equation are defined according to (2).

When μ_{v} is the Lebesgue measure, then the Lebesgue-Stieltjes integral of *f* is equivalent to the Lebesgue integral of *f*.

Where *f* is a real-valued function of a real variable and *v* is a non-decreasing real function, the Lebesgue-Stieltjes integral is equivalent to the Riemann-Stieltjes integral, in which case we often write

- Stanislaw Saks (1937).
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