Regulated integral
Encyclopedia
In mathematics
, the regulated integral is a definition of integration
for regulated function
s, which are defined to be uniform limits of step function
s. The use of the regulated integral instead of the Riemann integral
has been advocated by Nicolas Bourbaki
and Jean Dieudonné
.
, bounded
interval
in the real line
R. A real-valued function φ : [a, b] → R is called a step function if there exists a finite partition
of [a, b] such that φ is constant on each open
interval (ti, ti+1) of Π; suppose that this constant value is ci ∈ R. Then, define the integral of a step function φ to be
It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of [a, b] such that φ is constant on the open intervals of Π1, then the numerical value of the integral of φ is the same for Π1 as for Π.
if it is the uniform limit of a sequence of step functions on [a, b]:
Define the integral of a regulated function f to be
where (φn)n∈N is any sequence of step functions that converges uniformly to f.
One must check that this limit exists and is independent of the chosen sequence, but this
is an immediate consequence of the continuous linear extension theorem of elementary
functional analysis: a bounded linear operator T0 defined on a dense linear subspace
E0 of a normed linear space E and taking values in a Banach space F extends uniquely to a bounded linear operator T : E → F with the same (finite) operator norm
.
. However, care must be taken with certain technical points:
in the case of functions taking values in a normed vector space
X.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the regulated integral is a definition of integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
for regulated function
Regulated function
In mathematics, a regulated function is a "well-behaved" function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations...
s, which are defined to be uniform limits of step function
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
s. The use of the regulated integral instead of the Riemann integral
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. The Riemann integral is unsuitable for many theoretical purposes...
has been advocated by Nicolas Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
and Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
.
Definition on step functions
Let [a, b] be a fixed closedClosed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
, bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
in the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R. A real-valued function φ : [a, b] → R is called a step function if there exists a finite partition
Partition of an interval
In mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the formIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form...
of [a, b] such that φ is constant on each open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
interval (ti, ti+1) of Π; suppose that this constant value is ci ∈ R. Then, define the integral of a step function φ to be
It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of [a, b] such that φ is constant on the open intervals of Π1, then the numerical value of the integral of φ is the same for Π1 as for Π.
Extension to regulated functions
A function f : [a, b] → R is called a regulated functionRegulated function
In mathematics, a regulated function is a "well-behaved" function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations...
if it is the uniform limit of a sequence of step functions on [a, b]:
- there is a sequence of step functions (φn)n∈N such that || φn − f ||∞ → 0 as n → ∞; or, equivalently,
- for all ε > 0, there exists a step function φε such that || φε − f ||∞ < ε; or, equivalently,
- f lies in the closure of the space of step functions, where the closure is taken in the space of all bounded functionBounded functionIn mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
s [a, b] → R and with respect to the supremum norm || - ||∞; or equivalently, - for every t ∈ [a, b), the right-sided limit
-
- exists, and, for every t ∈ (a, b], the left-sided limit
- exists as well.
Define the integral of a regulated function f to be
where (φn)n∈N is any sequence of step functions that converges uniformly to f.
One must check that this limit exists and is independent of the chosen sequence, but this
is an immediate consequence of the continuous linear extension theorem of elementary
functional analysis: a bounded linear operator T0 defined on a dense linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
E0 of a normed linear space E and taking values in a Banach space F extends uniquely to a bounded linear operator T : E → F with the same (finite) operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
.
Properties of the regulated integral
- The integral is a linear operator: for any regulated functions f and g and constants α and β,
- The integral is also a bounded operatorBounded operatorIn functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
: every regulated function f is bounded, and if m ≤ f(t) ≤ M for all t ∈ [a, b], then
- In particular:
- Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
Extension to functions defined on the whole real line
It is possible to extend the definitions of step function and regulated function and the associated integrals to functions defined on the whole real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
. However, care must be taken with certain technical points:
- the partition on whose open intervals a step function is required to be constant is allowed to be a countable set, but must be a discrete set, i.e. have no limit pointLimit pointIn mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
s; - the requirement of uniform convergence must be loosened to the requirement of uniform convergence on compact setsCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, i.e. closedClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and boundedBounded setIn mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
intervals; - not every bounded functionBounded functionIn mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
is integrable (e.g. the function with constant value 1). This leads to a notion of local integrabilityLocally integrable functionIn mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.- Formal definition :...
.
Extension to vector-valued functions
The above definitions go through mutatis mutandisMutatis mutandis
Mutatis mutandis is a Latin phrase meaning "by changing those things which need to be changed" or more simply "the necessary changes having been made"....
in the case of functions taking values in a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
X.