Henstock-Kurzweil integral
Encyclopedia
In mathematics
, the Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is one of a number of definitions of the integral
of a function
. It is a generalization of the Riemann integral
which in some situations is more useful than the Lebesgue integral.
This integral was first defined by Arnaud Denjoy
(1912). Denjoy was interested in a definition that would allow one to integrate functions like
This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over [−ε,δ] and then let ε, δ → 0.
Trying to create a general theory Denjoy used transfinite induction
over the possible types of singularities which made the definition quite complicated. Other definitions were given by Nikolai Luzin
(using variations on the notions of absolute continuity
), and by Oskar Perron
, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
Later, in 1957, the Czech mathematician Jaroslav Kurzweil
discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock
. Due to these two important mathematicians, it is now commonly known as the Henstock–Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses, but this idea has not gained traction.
Given a tagged partition P of [a, b], say
and a positive function
which we call a gauge, we say P is
-fine if
For a tagged partition P and a function
we define the Riemann sum to be
Given a function
we now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge
such that whenever P is 
-fine, we have
If such an I exists, we say that f is Henstock–Kurzweil integrable on [a, b].
Cousin's theorem states that for every gauge
, such a 
-fine partition P does exist, so this condition cannot be satisfied vacuously
. The Riemann integral can be regarded as the special case where we only allow constant gauges.
If , then f is Henstock–Kurzweil integrable on [a, b] if and only if it is Henstock–Kurzweil integrable on both [a, c] and [c, b], and then
The Henstock–Kurzweil integral is linear, i.e., if f and g are integrable, and α, β are reals, then αf + βg is integrable and
If f is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and the values of the integrals are the same. The important Hake's theorem states that
whenever either side of the equation exists, and symmetrically for the lower integration bound. This means that if f is "improperly
Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals such as
are also Henstock–Kurzweil integrals. This shows that there is no sense in studying an "improper Henstock–Kurzweil integral" with finite bounds. However, it makes sense to consider improper Henstock–Kurzweil integrals with infinite bounds such as
For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if f is bounded, the following are equivalent:
In general, every Henstock–Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both f and |f| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent
version of Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem
(where the condition of dominance is loosened to g(x) ≤ fn(x) ≤ h(x) for some integrable g, h).
If F is differentiable everywhere (or with countable many exceptions), the derivative F′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is F. (Note that F′ need not be Lebesgue integrable.) In other words, we obtain a simpler and more satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative:
Conversely, the Lebesgue differentiation theorem
continues to holds for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and
then F′(x) = f(x) almost everywhere in [a, b] (in particular, F is almost everywhere differentiable).
The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm
, with respect to which it is barrelled
but incomplete
.
First of all, change of
to
(here
is a 
-neighbourhood of a) in the notion of 
-fine partition yields a definition of the Henstock–Kurzweil integral equivalent to the one given above. But after this change we can drop condition
and get a definition of McShane integral, which is equivalent to the Lebesgue integral.
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 Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is one of a number of definitions of the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. It is a generalization 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...
which in some situations is more useful than the Lebesgue integral.
This integral was first defined by Arnaud Denjoy
Arnaud Denjoy
Arnaud Denjoy was a French mathematician.Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives...
(1912). Denjoy was interested in a definition that would allow one to integrate functions like
This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over [−ε,δ] and then let ε, δ → 0.
Trying to create a general theory Denjoy used transfinite induction
Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinal numbers or cardinal numbers.- Transfinite induction :Let P be a property defined for all ordinals α...
over the possible types of singularities which made the definition quite complicated. Other definitions were given by Nikolai Luzin
Nikolai Luzin
Nikolai Nikolaevich Luzin, , was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the...
(using variations on the notions of absolute continuity
Absolute continuity
In mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity...
), and by Oskar Perron
Oskar Perron
Oskar Perron was a German mathematician.He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951...
, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
Later, in 1957, the Czech mathematician Jaroslav Kurzweil
Jaroslav Kurzweil
Jaroslav Kurzweil is a Czech mathematician. He is a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riemann sums...
discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock
Ralph Henstock
Ralph Henstock was an English mathematician and author. As an Integration theorist, he is notable for Henstock–Kurzweil integral...
. Due to these two important mathematicians, it is now commonly known as the Henstock–Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses, but this idea has not gained traction.
Definition
Henstock's definition is as follows:Given a tagged partition P of [a, b], say
and a positive function
which we call a gauge, we say P is
-fine ifFor a tagged partition P and a function
we define the Riemann sum to be
Given a function
we now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge
such that whenever P is -fine, we haveIf such an I exists, we say that f is Henstock–Kurzweil integrable on [a, b].
Cousin's theorem states that for every gauge
, such a -fine partition P does exist, so this condition cannot be satisfied vacuouslyVacuous truth
A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...
. The Riemann integral can be regarded as the special case where we only allow constant gauges.
Properties
Let be any function.If , then f is Henstock–Kurzweil integrable on [a, b] if and only if it is Henstock–Kurzweil integrable on both [a, c] and [c, b], and then
The Henstock–Kurzweil integral is linear, i.e., if f and g are integrable, and α, β are reals, then αf + βg is integrable and
If f is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and the values of the integrals are the same. The important Hake's theorem states that
whenever either side of the equation exists, and symmetrically for the lower integration bound. This means that if f is "improperly
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals such as
are also Henstock–Kurzweil integrals. This shows that there is no sense in studying an "improper Henstock–Kurzweil integral" with finite bounds. However, it makes sense to consider improper Henstock–Kurzweil integrals with infinite bounds such as
For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if f is bounded, the following are equivalent:
- f is Henstock–Kurzweil integrable,
- f is Lebesgue integrable,
- f is Lebesgue measurableMeasurable functionIn mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
.
In general, every Henstock–Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both f and |f| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
version of Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
(where the condition of dominance is loosened to g(x) ≤ fn(x) ≤ h(x) for some integrable g, h).
If F is differentiable everywhere (or with countable many exceptions), the derivative F′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is F. (Note that F′ need not be Lebesgue integrable.) In other words, we obtain a simpler and more satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative:
Conversely, the Lebesgue differentiation theorem
Lebesgue differentiation theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point...
continues to holds for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and
then F′(x) = f(x) almost everywhere in [a, b] (in particular, F is almost everywhere differentiable).
The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm
Alexiewicz norm
In mathematics — specifically, in integration theory — the Alexiewicz norm is an integral norm associated to the Henstock–Kurzweil integral. The Alexiewicz norm turns the space of Henstock–Kurzweil integrable functions into a topological vector space that is barrelled but...
, with respect to which it is barrelled
Barrelled space
In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
but incomplete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
.
McShane integral
Interestingly, Lebesgue integral on a line can also be presented in a similar fashion.First of all, change of
to
(here
is a -neighbourhood of a) in the notion of -fine partition yields a definition of the Henstock–Kurzweil integral equivalent to the one given above. But after this change we can drop conditionand get a definition of McShane integral, which is equivalent to the Lebesgue integral.
External links
The following are additional resources on the web for learning more:- http://www.math.vanderbilt.edu/~schectex/ccc/gauge/
- http://www.math.vanderbilt.edu/~schectex/ccc/gauge/letter/