Lebesgue–Stieltjes integration

Template:Short description In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes both Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.

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 is due. They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory.

Definition

The Lebesgue–Stieltjes integral

<math>\int_a^b f(x)\,dg(x)</math>

is defined when  <math>f : \left[a, b\right] \rightarrow \mathbb R</math>  is Borel-measurable and bounded and  <math>g : \left[a, b\right] \rightarrow \mathbb R</math>  is of bounded variation in Template:Math and right-continuous, or when Template:Math is non-negative and Template:Mvar is monotone and right-continuous. To start, assume that Template:Math is non-negative and Template:Mvar is monotone non-decreasing and right-continuous. Define Template:Math and Template:Math (Alternatively, the construction works for Template:Mvar left-continuous, Template:Math and Template:Math).

By Carathéodory's extension theorem, there is a unique Borel measure Template:Math on Template:Math which agrees with Template:Mvar on every interval Template:Mvar. The measure Template:Math arises from an outer measure (in fact, a metric outer measure) given by

<math>\mu_g(E) = \inf\left\{\sum_i \mu_g(I_i) \ : \ E\subseteq \bigcup_i I_i \right\}</math>

the infimum taken over all coverings of Template:Mvar by countably many semiopen intervals. This measure is sometimes called<ref>Halmos (1974), Sec. 15</ref> the Lebesgue–Stieltjes measure associated with Template:Mvar.

The Lebesgue–Stieltjes integral

<math>\int_a^b f(x)\,dg(x)</math>

is defined as the Lebesgue integral of Template:Math with respect to the measure Template:Math in the usual way. If Template:Mvar is non-increasing, then define

<math>\int_a^b f(x)\,dg(x) := -\int_a^b f(x) \,d (-g)(x),</math>

the latter integral being defined by the preceding construction.

If Template:Mvar is of bounded variation, then it is possible to write

<math>g(x)=g_1(x)-g_2(x)</math>

where Template:Math is the total variation of Template:Mvar in the interval Template:Math, and Template:Math. Both Template:Math and Template:Math are monotone non-decreasing.

Now, if Template:Math is bounded, the Lebesgue–Stieltjes integral of f with respect to Template:Mvar is defined by

<math>\int_a^b f(x)\,dg(x) = \int_a^b f(x)\,dg_1(x)-\int_a^b f(x)\,dg_2(x),</math>

where the latter two integrals are well-defined by the preceding construction.

Daniell integral

An alternative approach Template:Harv is to define the Lebesgue–Stieltjes integral as the Daniell integral that extends the usual Riemann–Stieltjes integral. Let Template:Mvar be a non-decreasing right-continuous function on Template:Math, and define Template:Math to be the Riemann–Stieltjes integral

<math>I(f) = \int_a^b f(x)\,dg(x)</math>

for all continuous functions Template:Math. The functional Template:Mvar defines a Radon measure on Template:Math. This functional can then be extended to the class of all non-negative functions by setting

<math>\begin{align}

\overline{I}(h) &= \sup \left \{I(f) \ : \ f\in C[a,b], 0\le f\le h \right \} \\ \overline{\overline{I}}(h) &= \inf \left \{I(f) \ : \ f \in C[a,b], h\le f \right \}. \end{align}</math>

For Borel measurable functions, one has

<math>\overline{I}(h) = \overline{\overline{I}}(h),</math>

and either side of the identity then defines the Lebesgue–Stieltjes integral of Template:Mvar. The outer measure Template:Math is defined via

<math>\mu_g(A) := \overline{I}(\chi_A)= \overline{\overline{I}}(\chi_A)</math>

where Template:Math is the indicator function of Template:Mvar.

Integrators of bounded variation are handled as above by decomposing into positive and negative variations.

Example

Suppose that Template:Math is a rectifiable curve in the plane and Template:Math is Borel measurable. Then we may define the length of Template:Mvar with respect to the Euclidean metric weighted by ρ to be

<math>\int_a^b \rho(\gamma(t))\,d\ell(t),</math>

where <math>\ell(t)</math> is the length of the restriction of Template:Mvar to Template:Math. This is sometimes called the Template:Mvar-length of Template:Mvar. This notion is quite useful for various applications: for example, in muddy terrain the speed in which a person can move may depend on how deep the mud is. If Template:Math denotes the inverse of the walking speed at or near Template:Mvar, then the Template:Mvar-length of Template:Mvar is the time it would take to traverse Template:Mvar. The concept of extremal length uses this notion of the Template:Mvar-length of curves and is useful in the study of conformal mappings.

Integration by parts

A function Template:Math is said to be "regular" at a point Template:Mvar if the right and left hand limits Template:Math and Template:Math exist, and the function takes at Template:Mvar the average value

<math>f(a)=\frac{f(a-)+f(a+)}{2}.</math>

Given two functions Template:Mvar and Template:Mvar of finite variation, if at each point either at least one of Template:Mvar or Template:Mvar is continuous or Template:Mvar and Template:Mvar are both regular, then an integration by parts formula for the Lebesgue–Stieltjes integral holds:<ref>Template:Cite journal</ref>

<math>\int_a^b U\,dV+\int_a^b V\,dU = U(b+)V(b+)-U(a-)V(a-), \qquad -\infty < a < b < \infty.</math>

Here the relevant Lebesgue–Stieltjes measures are associated with the right-continuous versions of the functions Template:Mvar and Template:Mvar; that is, to <math display="inline">\tilde U(x) = \lim_{t\to x^+} U(t)</math> and similarly <math>\tilde V(x).</math> The bounded interval Template:Open-open may be replaced with an unbounded interval Template:Open-open, Template:Open-open or Template:Open-open provided that Template:Mvar and Template:Mvar are of finite variation on this unbounded interval. Complex-valued functions may be used as well.

An alternative result, of significant importance in the theory of stochastic calculus is the following. Given two functions Template:Mvar and Template:Mvar of finite variation, which are both right-continuous and have left-limits (they are càdlàg functions) then

<math>U(t)V(t) = U(0)V(0) + \int_{(0,t]} U(s-)\,dV(s)+\int_{(0,t]} V(s-)\,dU(s)+\sum_{u\in (0,t]} \Delta U_u \Delta V_u,</math>

where Template:Math. This result can be seen as a precursor to Itô's lemma, and is of use in the general theory of stochastic integration. The final term is Template:Mathwhich arises from the quadratic covariation of Template:Mvar and Template:Mvar. (The earlier result can then be seen as a result pertaining to the Stratonovich integral.)

Related concepts

Lebesgue integration

When Template:Math for all real Template:Mvar, then Template:Math is the Lebesgue measure, and the Lebesgue–Stieltjes integral of Template:Math with respect to Template:Mvar is equivalent to the Lebesgue integral of Template:Math.

Riemann–Stieltjes integration and probability theory

Where Template:Math is a continuous real-valued function of a real variable and Template:Mvar is a non-decreasing real function, the Lebesgue–Stieltjes integral is equivalent to the Riemann–Stieltjes integral, in which case we often write

<math>\int_a^b f(x) \, dv(x)</math>

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

<math>\int_{-\infty}^\infty f(x) \, dv(x) = \mathrm{E}[f(X)].</math>

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

Also see

Henstock-Kurzweil-Stiltjes Integral

References

• Template:Citation
• Template:Citation.
• Saks, Stanisław (1937) Theory of the Integral.
• Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. Template:Isbn.