Lebesgue-Stieltjes integration

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.

Definition

The Lebesgue-Stieltjes integral $int\_a^b\; f(x),dg(x)$ is defined when$f:\; [a,b]\; oR$ is Borel-measurableand boundedand $g:\; [a,b]\; o\; R$ is of bounded variation in $[a,b]$ orwhen $f$ is non-negative and $g$ is monotone.To start, we assume that $f$ is non-negative and$g$ is monotone non-decreasing. In that case, for an interval $Isubset\; [a,b]$,define $w(I):=sup\_\{xin\; I\}g(x)-inf\_\{xin\; I\}g(x)$ (this is just $g(t)-g(s)$ if $I=\; [s,t]$, but we allow $I$ to be not necessarily closed).By Carathéodory's extension theorem, there is a unique Borel measure $mu\_g$ on$[a,b]$ which agrees with $w$ on every interval $I$.This measure is sometimes called [Halmos (1974), Sec. 15] the Lebesgue-Stieltjes measure associated with $g$.Then $int\_a^b\; f(x),dg(x)$ is defined as the Lebesgue integral of $f$ with respect to the measure $mu\_g$.If $g$ is non-increasing, then $int\_a^b\; f(x),dg(x)$ is defined as$-int\_a^b\; f(x)\; ,d\; (-g)(x)$.

If $g$ is of bounded variation and $f$ is bounded, then we may write$g(x)=g\_1(x)-g\_2(x),$where $g\_1(x):=V\_a^xg$ is the total variationof $g$ in the interval $[a,x]$, and $g\_2(x)=g\_1(x)-g(x)$.It can easily be shown that $g\_1$ and $g\_2$ are both monotone non-decreasing.Now the Lebesgue-Stieltjes integral $int\_a^b\; f(x),dg(x)$ is defined as$int\_a^b\; f(x),dg\_1(x)-int\_a^b\; f(x),dg\_2(x)$, where the latter two integralsare well defined since $g\_1$ and $g\_2$ are non-decreasing.

Example

Suppose that $gamma:\; [a,b]\; oR^2$ is a rectifiable curve in the planeand $ho:R^2\; o\; [0,infty)$ is Borel measurable. Then we may define the lengthof $gamma$ with respect to the Euclidean metric weighted by $ho$ tobe $int\_a^b\; ho(gamma(t)),dell(t)$, where $ell(t)$ is the lengthof the restriction of $gamma$ to $[a,t]$.This is sometimes called the $ho$-length of $gamma$.This notion is quite useful forvarious applications: for example, in muddy terrain the speed in which a person can move maydepend on how deep the mud is. If $ho(z)$ denotes the inverse of the walking speedat or near $z$, then the $ho$-length of $gamma$ is thetime it would take to traverse $gamma$. The concept of extremal length usesthis notion of the $ho$-length of curves and is useful in the study of
conformal mappings.

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,:$f(a)=frac\{1\}\{2\}left(f(a-)+f(a+)\; ight)$,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 $b>a$. Under a slight generalization of this formula, the extra conditions on $U$ and $V$ can be dropped. [cite journal |last=Hewitt |first=Edwin |year=1960 |month=5 |title=Integration by Parts for Stieltjes Integrals |journal=The American Mathematical Monthly |volume=67 |issue=5 |pages=419–423 |url=http://www.jstor.org/pss/2309287 |accessdate= 2008-04-23 |doi=10.2307/2309287 ]

Related concepts

Lebesgue integration

When μ_"v" 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 continuous 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 :$int\_\{-infty\}^infty\; f(x)\; ,\; dv(x)\; =\; mathrm\{E\}\; [f(X)]\; .$(See the article on Riemann-Stieltjes integration for more detail on dealing with such cases.)

Notes

References

*
* 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.

External links

* Saks, Stanislaw (1937) " [http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10 Theory of the Integral.] "
* [http://www.probability.net/ www.probability.net Probability and foundations tutorial.]