Lebesgue-Stieltjes integration

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In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes 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.

1 Formal construction
2 Integration by parts
3 Related concepts
- 3.1 Lebesgue integration
- 3.2 Riemann-Stieltjes integration and probability theory
4 Reference
5 External links

[edit] Formal construction

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 w has support on F, then define

$(1) \quad \mu_w(E) := \inf \left\{\sum_j w(I_j) : E \subseteq \Omega, \, E \subset \bigcup_j I_j \right\},$

(the lower bound over all sequences of intervals {I_j}). Note that it is possible to show that μ_w 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, μ_w) is a measure space, then we can define the integral of any simple function s = Σ_i a_i1_Ai (where 1_A is the indicator function of A) as

$\int s \, d\mu_w = \sum_i a_i \mu_w(A_i).$

Then, if f is a μ_w-measurable map, f:(Ω, F) → [0, +∞], we can define the integral of f with respect to μ_w over E ⊆ Ω, as

$(2) \quad \int_E f \, d\mu_w = \sup\left\{\int s\,d\mu_w^E : s < f, s\ \mbox{simple}\,\right\},$

where μ_w^E(·) = μ_w(E∩·) on E and 0 otherwise. (If E = Ω, μ_w^Ω = μ_w.)

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

$\int_E f \, d\mu_w = \int_E g \, d\mu_w - \int_E h \, d\mu_w.$

We now have a theory of Lebesgue-Stieltjes integrals of arbitrary functions f, with respect to measures μ_w associated with non-negative additive functions of an interval, of bounded variation. We generally want to deal with measures associated with arbitrary additive functions, however, 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

$(3) \quad \mu_v(E) = \mu_{w_1}(E) - \mu_{-w_2}(E),$

where the measures μ_w1 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 μ_v is defined according to equations (1) and (3), the Lebesgue-Stieltjes integral of f with respect to μ_v is

$\int_E f \, d\mu_v = \left(\int_E g \, d\mu_{w_1} - \int_E h \, d\mu_{w_1}\right) - \left(\int_E g \, d\mu_{-w_2} - \int_E h \, d\mu_{-w_2}\right),$

where each of the integrals on the right hand side of this equation are defined according to (2).

[edit] Integration by parts

A function $f$ is said to be "regular" at a point $a$ if the right and left hand limits $f (a + )$ and $f (a - )$ exist, and the function takes the average value,

$f(a)=\frac{1}{2}\left(f(a-)+f(a+)\right)$ ,

at the limiting point. Given two functions $U$ and $V$ , if at each point either $U$ or $V$ is continuous, or if both $U$ and $V$ are regular, then there is an integration by parts formula for the Lebesgue-Stieltjes integral:

$\int_a^b U\,dV+\int_a^b V\,dU=U(b+)V(b+)-U(a-)V(a-)$ ,

where $b > a$ .

[edit] Related concepts

[edit] Lebesgue integration

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

[edit] 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

$\int_a^b f(x) \, dv(x)$

for the Lebesgue-Stieltjes integral, letting the measure μ_v remain implicit. This is particularly common in probability theory when v is the cumulative distribution function of a real-valued random variable, in which case

$\int_0^1 f(x) \, dv(x) = \mathrm{E}[f(X)].$

(See the article on Riemann-Stieltjes integration for more detail on dealing with such cases.)

[edit] Reference

Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.