Riemann–Stieltjes Integral
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann–Stieltjes integral is a generalization of the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
, named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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 at ...
. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the
Lebesgue integral 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 Lebe ...
, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.


Formal definition

The Riemann–Stieltjes
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real fun ...
f of a real variable on the interval ,b/math> with respect to another real-to-real function g is denoted by :\int_^b f(x) \, \mathrmg(x). Its definition uses a sequence of
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
P of the interval ,b/math> :P=\. The integral, then, is defined to be the limit, as the
mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, ex ...
(the length of the longest subinterval) of the partitions approaches 0 , of the approximating sum :S(P,f,g) = \sum_^ f(c_i)\left g(x_) - g(x_i) \right/math> where c_i is in the i-th subinterval _i;x_/math>. The two functions f and g are respectively called the ''integrand'' and the ''integrator''. Typically g is taken to be
monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
(or at least of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
) and right-semicontinuous (however this last is essentially convention). We specifically do not require g to be continuous, which allows for integrals that have point mass terms. The "limit" is here understood to be a number ''A'' (the value of the Riemann–Stieltjes integral) such that for every ''ε'' > 0, there exists ''δ'' > 0 such that for every partition ''P'' with norm(''P'') < ''δ'', and for every choice of points ''c''''i'' in 'x''''i'', ''x''''i''+1 :, S(P,f,g)-A, < \varepsilon. \,


Properties

The Riemann–Stieltjes integral admits
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 antiderivative. ...
in the form :\int_a^b f(x) \, \mathrmg(x)=f(b)g(b)-f(a)g(a)-\int_a^b g(x) \, \mathrmf(x) and the existence of either integral implies the existence of the other. On the other hand, a classical result shows that the integral is well-defined if ''f'' is ''α''-
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
and ''g'' is ''β''-Hölder continuous with  . If f(x) is bounded on ,b/math>, g(x) increases monotonically, and g'(x) is Riemann integrable, then the Riemann–Stieltjes integral is related to the Riemann integral by \int_a^b f(x) \, \mathrmg(x) = \int_a^b f(x) g'(x) \, \mathrmx For a step function g(x) = \begin 0 & \text x \leq s \\ 1 & \text x > s \\ \end where a < s < b, if f is continuous at s, then \int_a^b f(x) \, \mathrmg(x) = f(s)


Application to probability theory

If ''g'' is the
cumulative probability 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
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' that has a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
with respect to
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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
, and ''f'' is any function for which the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
\operatorname\left f(X)\\,\right/math> is finite, then the probability density function of ''X'' is the derivative of ''g'' and we have :\operatorname
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
\int_^\infty f(x)g'(x)\,\mathrmx. But this formula does not work if ''X'' does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of ''X'' is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function ''g'' is continuous, it does not work if ''g'' fails to be
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
(again, the
Cantor function In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Th ...
may serve as an example of this failure). But the identity :\operatorname
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
\int_^\infty f(x)\, \mathrmg(x) holds if ''g'' is ''any'' cumulative probability distribution function on the real line, no matter how ill-behaved. In particular, no matter how ill-behaved the cumulative distribution function ''g'' of a random variable ''X'', if the moment E(''X''''n'') exists, then it is equal to : \operatorname\left ^n\right= \int_^\infty x^n\,\mathrmg(x).


Application to functional analysis

The Riemann–Stieltjes integral appears in the original formulation of
F. Riesz's theorem F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences ...
which represents the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of the
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
''C'' 'a'',''b''of continuous functions in an interval 'a'',''b''as Riemann–Stieltjes integrals against functions of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
. Later, that theorem was reformulated in terms of measures. The Riemann–Stieltjes integral also appears in the formulation of the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
for (non-compact) self-adjoint (or more generally, normal) operators in a Hilbert space. In this theorem, the integral is considered with respect to a spectral family of projections.


Existence of the integral

The best simple existence theorem states that if ''f'' is continuous and ''g'' is of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
on 'a'', ''b'' then the integral exists. A function ''g'' is of bounded variation if and only if it is the difference between two (bounded) monotone functions. If ''g'' is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to ''g''. In general, the integral is not well-defined if ''f'' and ''g'' share any points of discontinuity, but there are other cases as well.


Geometric Interpretation

A 3D plot, with x, f(x), and g(x) all along orthogonal axes, leads to a geometric interpretation of the Riemann–Stieltjes integral. If the g-x plane is horizontal and the f-direction is pointing upward, then the surface to be considered is like a curved fence. The fence follows the curve traced by g(x), and the height of the fence is given by f(x). The fence is the section of the g-sheet (i.e., the g curve extended along the f axis) that is bounded between the g-x plane and the f-sheet. The Riemann-Stieljes integral is the area of the projection of this fence onto the f-g plane — in effect, its "shadow". The slope of g(x) weights the area of the projection. The values of x for which g(x) has the steepest slope g'(x) correspond to regions of the fence with the greater projection and thereby carry the most weight in the integral. When g is a step function g(x) = \begin 0 & \text x \leq s \\ 1 & \text x > s \\ \end the fence has a rectangular "gate" of width 1 and height equal to f(s). Thus the gate, and its projection, have area equal to f(s), the value of the Riemann-Stieljes integral.


Generalization

An important generalization is the Lebesgue–Stieltjes integral, which generalizes the Riemann–Stieltjes integral in a way analogous to how the
Lebesgue integral 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 Lebe ...
generalizes the Riemann integral. If improper Riemann–Stieltjes integrals are allowed, then the Lebesgue integral is not strictly more general than the Riemann–Stieltjes integral. The Riemann–Stieltjes integral also generalizes to the case when either the integrand ''ƒ'' or the integrator ''g'' take values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. If takes values in the Banach space ''X'', then it is natural to assume that it is of strongly bounded variation, meaning that :\sup \sum_i \, g(t_)-g(t_i)\, _X < \infty the supremum being taken over all finite partitions :a=t_0\le t_1\le\cdots\le t_n=b of the interval 'a'',''b'' This generalization plays a role in the study of
semigroups In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
, via the
Laplace–Stieltjes transform The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is o ...
. The Itô integral extends the Riemann–Stietjes integral to encompass integrands and integrators which are
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. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es rather than simple functions; see also
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 ...
.


Generalized Riemann–Stieltjes integral

A slight generalization is to consider in the above definition partitions ''P'' that ''refine'' another partition ''P''''ε'', meaning that ''P'' arises from ''P''''ε'' by the addition of points, rather than from partitions with a finer mesh. Specifically, the generalized Riemann–Stieltjes integral of ''f'' with respect to ''g'' is a number ''A'' such that for every ''ε'' > 0 there exists a partition ''P''''ε'' such that for every partition ''P'' that refines ''P''''ε'', :, S(P,f,g) - A, < \varepsilon \, for every choice of points ''c''''i'' in 'x''''i'', ''x''''i''+1 This generalization exhibits the Riemann–Stieltjes integral as the Moore–Smith limit on the
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
of partitions of 'a'', ''b''nbsp;. A consequence is that with this definition, the integral \int_a^b f(x)\,\mathrmg(x) can still be defined in cases where ''f'' and ''g'' have a point of discontinuity in common.


Darboux sums

The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition ''P'' and a nondecreasing function ''g'' on 'a'', ''b''define the upper Darboux sum of ''f'' with respect to ''g'' by :U(P,f,g) = \sum_^n \,\, ,g(x_i)-g(x_)\,\,\sup_ f(x) and the lower sum by :L(P,f,g) = \sum_^n \,\, ,g(x_i)-g(x_)\,\,\inf_ f(x). Then the generalized Riemann–Stieltjes of ''f'' with respect to ''g'' exists if and only if, for every ε > 0, there exists a partition ''P'' such that :U(P,f,g)-L(P,f,g) < \varepsilon. Furthermore, ''f'' is Riemann–Stieltjes integrable with respect to ''g'' (in the classical sense) if :\lim_ ,U(P,f,g)-L(P,f,g)\,= 0.\quad


Examples and special cases


Differentiable ''g''(''x'')

Given a g(x) which is continuously
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
over \mathbb it can be shown that there is the equality : \int_a^b f(x) \, \mathrmg(x) = \int_a^b f(x)g'(x) \, \mathrmx where the integral on the right-hand side is the standard Riemann integral, assuming that f can be integrated by the Riemann–Stieltjes integral. More generally, the Riemann integral equals the Riemann–Stieltjes integral if g is the
Lebesgue integral 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 Lebe ...
of its derivative; in this case g is said to be
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
. It may be the case that g has jump discontinuities, or may have derivative zero ''almost'' everywhere while still being continuous and increasing (for example, g could be the
Cantor function In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Th ...
or “Devil's staircase”), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of ''g''.


Riemann integral

The standard Riemann integral is a special case of the Riemann–Stieltjes integral where g(x) = x.


Rectifier

Consider the function g(x) = \max\ used in the study of
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
s, called a ''rectified linear unit'' (ReLU). Then the Riemann–Stieltjes can be evaluated as : \int_a^b f(x)\,\mathrmg(x) = \int_^f(x)\,\mathrmx where the integral on the right-hand side is the standard Riemann integral.


Cavalieri integration

Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
can be used to calculate areas bounded by curves using Riemann–Stieltjes integrals.T. L. Grobler, E. R. Ackermann, A. J. van Zyl & J. C. Olivier
Cavaliere integration
from
Council for Scientific and Industrial Research The Council for Scientific and Industrial Research (CSIR) is South Africa's central and premier scientific research and development organisation. It was established by an act of parliament in 1945 and is situated on its own campus in the cit ...
The integration strips of Riemann integration are replaced with strips that are non-rectangular in shape. The method is to transform a "Cavaliere region" with a transformation h, or to use g = h^ as integrand. For a given function f(x) on an interval ,b/math>, a "translational function" a(y) must intersect (x,f(x )) exactly once for any shift in the interval. A "Cavaliere region" is then bounded by f(x),a(y), the x-axis, and b(y) = a(y) + (b-a). The area of the region is then :\int_^ f(x) \, dx \ = \ \int_^ f(x) \, dg(x) , where a' and b' are the x-values where a(y) and b(y) intersect f(x).


Notes


References

* * via
HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digitized locally ...
* * * * * * * * * * * * {{DEFAULTSORT:Riemann-Stieltjes integral Definitions of mathematical integration Bernhard Riemann