HOME

TheInfoList



OR:

In mathematics, the regulated integral is a definition of
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
for
regulated function In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated ...
s, which are defined to be uniform limits of
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. Informally speaking, a step function is a piecewise constant function having onl ...
s. The use of the regulated integral instead 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 G� ...
has been advocated by
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
and
Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous ...
.


Definition


Definition on step functions

Let 'a'', ''b''be a fixed
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
, bounded interval in the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
R. A real-valued function ''φ'' : 'a'', ''b''→ R is called a step function if there exists a finite partition :\Pi = \ of 'a'', ''b''such that ''φ'' is constant on each
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
interval (''t''''i'', ''t''''i''+1) of Π; suppose that this constant value is ''c''''i'' ∈ R. Then, define the integral of a step function ''φ'' to be :\int_a^b \varphi(t) \, \mathrm t := \sum_^ c_i , t_ - t_i , . 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 function In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated ...
if it is the uniform limit of a sequence of step functions on 'a'', ''b'' * there is a sequence of step functions (''φ''''n'')''n''∈N such that as ''n'' → ∞; or, equivalently, * for all ''ε'' > 0, there exists a step function ''φ''''ε'' such that , , ''φ''''ε'' − ''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 function In 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'' such that :, f(x), \le M for all ''x'' in ''X''. A ...
s 'a'', ''b''→ R and with respect to the
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
, ,  ⋅ , , ; or equivalently, * for every , the right-sided limit *::f(t+) = \lim_ f(s) *:exists, and, for every , the left-sided limit *::f(t-) = \lim_ f(s) *:exists as well. Define the integral of a regulated function ''f'' to be :\int_^ f(t) \, \mathrm t := \lim_ \int_^ \varphi_ (t) \, \mathrm t, where (''φ''''n'')''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 In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X by first defining a linear transformation \mathsf on a dense subset of X and then extending \mathsf to the whole space via the t ...
theorem of elementary functional analysis: a bounded linear operator ''T''0 defined on a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
''E''0 of a
normed linear space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
''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 In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Intro ...
.


Properties of the regulated integral

* The integral is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
: for any regulated functions ''f'' and ''g'' and constants ''α'' and ''β'', *::\int_^ \alpha f(t) + \beta g(t) \, \mathrm t = \alpha \int_^ f(t) \, \mathrm t + \beta \int_^ g(t) \, \mathrm t. * The integral is also a bounded operator: every regulated function ''f'' is bounded, and if ''m'' ≤ ''f''(''t'') ≤ ''M'' for all ''t'' ∈ 'a'', ''b'' then *::m , b - a , \leq \int_^ f(t) \, \mathrm t \leq M , b - a , . *: In particular: *::\left, \int_^ f(t) \, \mathrm t \ \leq \int_^ , f(t) , \, \mathrm t. * 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 line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
. 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 ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
, i.e. have no
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point 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 conta ...
s; * the requirement of uniform convergence must be loosened to the requirement of uniform convergence on compact sets, i.e.
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
and bounded intervals; * not every
bounded function In 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'' such that :, f(x), \le M for all ''x'' in ''X''. A ...
is integrable (e.g. the function with constant value 1). This leads to a notion of local integrability.


Extension to vector-valued functions

The above definitions go through ''
mutatis mutandis ''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...
'' in the case of functions taking values in a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
''X''.


See also

*
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 Le ...
*
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 G� ...


References

* * {{Functional Analysis Definitions of mathematical integration