In
measure-theoretic analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and related branches of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Lebesgue–Stieltjes integration generalizes both
Riemann–Stieltjes and
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis. The Lebesgue integral, named after French mathematician Henri L ...
, 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
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
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
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s, named for
Henri Leon Lebesgue and
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes ( , ; 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics ...
, are also known as Lebesgue–Radon integrals or just Radon integrals, after
Johann Radon
Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna).
Life
RadonBrigitte Bukovics: ''Biography of Johan ...
, to whom much of the theory is due. They find common application in
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
and
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es, and in certain branches of
analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
including
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
.
Definition
The Lebesgue–Stieltjes integral
:
is defined when
is
Borel-
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
and
bounded and
is of
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
in and right-continuous, or when is non-negative and is
monotone and
right-continuous
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
. To start, assume that is non-negative and is monotone non-decreasing and right-continuous. Define and (Alternatively, the construction works for left-continuous, and ).
By
Carathéodory's extension theorem, there is a unique Borel measure on which agrees with on every interval . The measure arises from an
outer measure
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer me ...
(in fact, a
metric outer measure) given by
:
the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
taken over all coverings of by countably many semiopen intervals. This measure is sometimes called the Lebesgue–Stieltjes measure associated with .
The Lebesgue–Stieltjes integral
:
is defined as the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
of with respect to the measure in the usual way. If is non-increasing, then define
:
the latter integral being defined by the preceding construction.
If is of bounded variation, then it is possible to write
:
where is the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real ...
of in the interval , and . Both and are monotone non-decreasing.
Now, if is bounded, the Lebesgue–Stieltjes integral of f with respect to is defined by
:
where the latter two integrals are well-defined by the preceding construction.
Daniell integral
An alternative approach is to define the Lebesgue–Stieltjes integral as the
Daniell integral that extends the usual Riemann–Stieltjes integral. Let be a non-decreasing right-continuous function on , and define to be the Riemann–Stieltjes integral
:
for all continuous functions . The
functional defines a
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and ...
on . This functional can then be extended to the class of all non-negative functions by setting
:
For Borel measurable functions, one has
:
and either side of the identity then defines the Lebesgue–Stieltjes integral of . The outer measure is defined via
:
where is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of .
Integrators of bounded variation are handled as above by decomposing into positive and negative variations.
Example
Suppose that is a
rectifiable curve
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
in the plane and is Borel measurable. Then we may define the length of with respect to the Euclidean metric weighted by ρ to be
:
where
is the length of the restriction of to . This is sometimes called the -length of . 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 denotes the inverse of the walking speed at or near , then the -length of is the time it would take to traverse . The concept of
extremal length uses this notion of the -length of curves and is useful in the study of
conformal map
In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
pings.
Integration by parts
A function is said to be "regular" at a point if the right and left hand limits and exist, and the function takes at the average value
:
Given two functions and of finite variation, if at each point either at least one of or is continuous or and are both regular, then an
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
formula for the Lebesgue–Stieltjes integral holds:
:
Here the relevant Lebesgue–Stieltjes measures are associated with the right-continuous versions of the functions and ; that is, to
and similarly
The bounded interval may be replaced with an unbounded interval , or provided that and 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
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
is the following. Given two functions and of finite variation, which are both right-continuous and have left-limits (they are
càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous an ...
functions) then
:
where . This result can be seen as a precursor to
Itô's lemma
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
, and is of use in the general theory of stochastic integration. The final term is which arises from the quadratic covariation of and . (The earlier result can then be seen as a result pertaining to the
Stratonovich integral
In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô in ...
.)
Related concepts
Lebesgue integration
When for all real , then is the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
, and the Lebesgue–Stieltjes integral of with respect to is equivalent to the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
of .
Riemann–Stieltjes integration and probability theory
Where is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
real-valued function of a real variable and is a non-decreasing real function, the Lebesgue–Stieltjes integral is equivalent to the
Riemann–Stieltjes integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, in which case we often write
:
for the Lebesgue–Stieltjes integral, letting the measure remain implicit. This is particularly common in
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
when is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...
of a real-valued random variable , in which case
:
(See the article on
Riemann–Stieltjes integration for more detail on dealing with such cases.)
Notes
Also see
Henstock-Kurzweil-Stiltjes Integral
References
*
*.
*
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. .
{{DEFAULTSORT:Lebesgue-Stieltjes integration
Definitions of mathematical integration