In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Riemann–Stieltjes integral is a generalization of the
Riemann integralIn 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. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define...
, named after
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
and
Thomas Joannes StieltjesThomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France...
.
The Riemann–Stieltjes integral of a
realIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
-valued function
f of a real variable with respect to a real function
g is denoted by
and defined to be the limit, as the
meshIn mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the formSuch partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral...
of the
partitionIn mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the formSuch partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral...
of the interval [
a,
b] approaches zero, of the approximating sum
where
ci is in the
i-th subinterval [
xi,
xi+1].
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Riemann–Stieltjes integral is a generalization of the
Riemann integralIn 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. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define...
, named after
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
and
Thomas Joannes StieltjesThomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France...
.
Definition
The Riemann–Stieltjes integral of a
realIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
-valued function
f of a real variable with respect to a real function
g is denoted by
and defined to be the limit, as the
meshIn mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the formSuch partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral...
of the
partitionIn mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the formSuch partitions are used in the theory of the Riemann integral, the Riemann-Stieltjes integral and the regulated integral...
of the interval [
a,
b] approaches zero, of the approximating sum
where
ci is in the
i-th subinterval [
xi,
xi+1]. The two functions
f and
g are respectively called the integrand and the integrator. Most commonly,
g will be nondecreasing, but this is not required. For this Riemann–Stieltjes integral to exist,
f and
g must not share any points of discontinuity. (If
g is continuous, but not of
bounded variationIn mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
, then too the integral may not exist, hence the discontinuity condition is not sufficient.)
An alternative, and slightly more general, definition of the Riemann–Stieltjes integral uses the same approximating sums as above, but takes the limit to be a Moore–Smith limit on the directed set of partitions of [
a,
b]. That is, take the limit as more and more division points are inserted into the partition. With this definition, an integral can exist when
f and
g share points of discontinuity, as long as they are not discontinuous from the same side at the same point.
For another formulation that is much more general, see the Lebesgue–Stieltjes integral. It is notable however, that if improper Riemann–Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general.
Properties and relation to the Riemann integral
If
g should happen to be everywhere differentiable, then the integral may still be different from the
Riemann integralIn 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. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define...
for example, if the derivative is unbounded. But if the derivative is continuous, they will be the same. This condition is also satisfied if
g is the (Lebesgue) integral of its derivative; in this case
g is said to be absolutely continuous.
However,
g may have jump discontinuities, or may have derivative zero
almost everywhere while still being continuous and increasing (for example,
g could be the
Cantor functionIn mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:...
or
Minkowski's question mark functionIn mathematics, the Minkowski question mark function, sometimes called the slippery devil's staircase and denoted by ?, is a function possessing various unusual fractal properties, defined by Hermann Minkowski in 1904...
), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of
g.
The Riemann–Stieltjes integral admits
integration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals...
in the form
and the existence of the integral on the left implies the existence of the integral on the right.
Existence of the integral
The best simple existence theorem states that if
f is continuous and
g is of
bounded variationIn mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
on [
a,
b], then the integral exists. Note that
g is of bounded variation if and only if it is the difference between two monotone functions. If
g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to
g.
Application to probability theory
If
g is the
cumulative probability distribution functionIn probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X...
of a
random variableIn mathematics, random variables are used in the study of probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions...
X that has a
probability density functionIn probability theory, a probability density function —often referred to as a probability distribution function—or density, of a random variable is a function that describes the density of probability at each point in the sample space...
with respect to
Lebesgue measureIn mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration...
, and
f is any function for which the
expected valueIn probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure....
E(|
f(
X)|) is finite, then the probability density function of
X is the derivative of
g and we have
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 continuousIn mathematics, absolute continuity is a smoothness property which is stricter than continuity and uniform continuity. Both absolute continuity of functions and absolute continuity of measures are defined.-Definition:...
(again, the
Cantor functionIn mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:...
may serve as an example of this failure). But the identity
holds if
g is
any cumulative probability distribution function on the real line, no matter how ill-behaved.
Application to functional analysis
The Riemann–Stieltjes integral appears in the original formulation of F. Riesz's theorem which represents the
dual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of the
Banach spaceIn mathematics, Banach spaces are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions , spaces of Lebesgue integrable functions known as Lp spaces,...
of continuous functions in an interval as Riemann–Stieltjes integrals against functions of
bounded variationIn mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
(later, that theorem was reformulated in terms of measures).
Also, the Riemann–Stieltjes integral appears in the formulation of the
spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
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 so-called spectral family of projections). See Reisz 1955 for details.
Generalization
An important generalization is the Lebesgue–Stieltjes integral which generalizes the Riemann–Stieltjes integral in a way analogous to how the Lebesgue integral generalizes the Riemann integral.
The Riemann–Stieltjes integral also generalizes to the case when either the integrand
ƒ or the integrator
g take values in a
Banach spaceIn mathematics, Banach spaces are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions , spaces of Lebesgue integrable functions known as Lp spaces,...
. If takes values in the Banach space
X, then it is natural to assume that it is of
strongly bounded variation, meaning that
the supremum being taken over all finite partitions
of the interval [
a,
b]. This generalization plays a role in the study of
semigroupsIn mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function...
, via the Laplace–Stieltjes transform.