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Lebesgue-Stieltjes integration

If you are having difficulty understanding this article, you might want to first learn more about integrals, particularly the Lebesgue integral, and measure theory.

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.

Table of contents
1 Formal construction
2 Related concepts
3 External links

Formal construction

In order to define the Lebesgue-Stieltjes integral, we will begin by associating a measure, μw, 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 w has support on F, then define

(the lower bound over all sequences of intervals {Ij}). Note that it is possible to show that μw is an outer measure.

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, μw) is a measure space, then we can define the integral of any simple function s = Σi ai1Ai (where 1A is the indicator function of A) as

Then, if f is a μw-measurable map, f:(Ω, F) → [0, +∞], we can define the integral of f with respect to μw over E ⊆ Ω, as
where μwE(·) = μw(E∩·) on E and 0 otherwise. (If E = Ω, μwΩ = μw.)

It is often required, of course, to compute the integral of arbitrary measurable functions f:(Ω, F) → R∪{-∞, +∞}, but (as for the Lebesgue integral) we may construct these from two non-negative functions. If g:(Ω, F) → [0, +∞] and h:(Ω, F) → [0, +∞] such that g = max(0,f) and h = max(-f,0), then clearly f = g - h and

We now have a theory of Lebesgue-Stieltjes integrals of arbitrary functions f, with respect to measures μw associated with non-negative additive functions of an interval, of bounded variation. However, we generally want to deal with measures associated with arbitrary additive functions, however, and so suppose that v is an arbitrary (i.e. possibly not non-negative) additive function of an interval, again of bounded variation. Let w1 and w2 denote the upper and lower variations of v, respectively. Then
where the measures μw1 and μ-w2 are defined as in equation (1), above.

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.

Let g = max(0, f) and h = max(-f, 0), and let w1 and w2 be the upper and lower variations of v, respectively. Then if μv is defined according to equations (1) and (3), the Lebesgue-Stieltjes integral of f with respect to μv is
where each of the integrals on the right hand side of this equation are defined according to (2).

Related concepts

Lebesgue integration

When μv is the Lebesgue measure, then the Lebesgue-Stieltjes integral of f is equivalent to the Lebesgue integral of f.

Riemann-Stieltjes integration and probability theory

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

for the Lebesgue-Stieltjes integral, letting the measure μv remain implicit. This is particularly common in probability theory when v is the distribution function of a real-valued random variable, in which case
(See the article on Riemann-Stieltjes integration for more detail on dealing with such cases.)

External links