Henstock–Kurzweil integral
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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 Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the
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
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
. It 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 ...
, and in some situations is more general than 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 ...
. In particular, a function is Lebesgue integrable if and only if the function and its
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
are Henstock–Kurzweil integrable. This integral was first defined by
Arnaud Denjoy Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician. Biography Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. Henstock–Kurzweil integral, His integral ...
(1912). Denjoy was interested in a definition that would allow one to integrate functions like :f(x)=\frac\sin\left(\frac\right). This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval and then let . Trying to create a general theory, Denjoy used
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...
over the possible types of singularities, which made the definition quite complicated. Other definitions were given by
Nikolai Luzin Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...
(using variations on the notions of
absolute continuity 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 ope ...
), 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 Jaroslav Kurzweil (, 7 May 1926, Prague – 17 March 2022) was a Czech mathematician. Biography Born in Prague, Czechoslovakia, he was a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riema ...
discovered a new definition of this integral elegantly similar in nature to
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 ...
's original definition which he named the gauge integral.
Ralph Henstock Ralph Henstock (2 June 1923 – 17 January 2007) was an English mathematician and author. As an Integral#Other integrals, Integration theorist, he is notable for Henstock–Kurzweil integral. Henstock brought the theory to a highly developed stage ...
independently introduced a similar integral that extended the theory in 1961, citing his investigations of Ward's extensions to the Perron integral. Due to these two important contributions, 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.


Definition

Given a tagged partition of , that is, :a = u_0 < u_1 < \cdots < u_n = b together with each partition's tag defined as a point :t_i \in _, u_i we define the Riemann sum for a function f \colon , b\to \mathbb to be : \sum_P f = \sum_^n f(t_i) \Delta u_i. where \Delta u_i := u_i - u_. This is the summation of each partition's length (\Delta u_i) multiplied by the function evaluated at that partition's tag (f(t_i)). Given a positive function :\delta \colon , b\to (0, \infty), which we call a ''gauge'', we say a tagged partition ''P ''is \delta-fine if :\forall i \ \ _, u_i\subset _i-\delta(t_i), t_i + \delta (t_i) We now define a number to be the Henstock–Kurzweil integral of if for every there exists a gauge \delta such that whenever is \delta-fine, we have : \left \vert I - \sum_P f \right \vert < \varepsilon. If such an exists, we say that is Henstock–Kurzweil integrable on . Cousin's theorem states that for every gauge \delta, such a \delta-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.


Properties

Let be any function. Given , is Henstock–Kurzweil integrable on if and only if it is Henstock–Kurzweil integrable on both and ; in which case, :\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx. Henstock–Kurzweil integrals are linear. Given integrable functions , and real numbers , , the expression is integrable; for example, :\int_a^b \left(\alpha f(x) + \beta g(x)\right) dx = \alpha \int_a^bf(x)\,dx + \beta \int_a^b g(x)\,dx. If ''f'' is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and calculating that integral gives the same result by all three formulations. The important Hake's theorem states that :\int_a^b f(x)\,dx = \lim_ \int_a^c f(x)\,dx whenever either side of the equation exists, and likewise symmetrically for the lower integration bound. This means that if is " improperly Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals of types such as :\int_0^1 \fracx\,dx are also proper Henstock–Kurzweil integrals. To study an "improper Henstock–Kurzweil integral" with finite bounds would not be meaningful. However, it does make sense to consider improper Henstock–Kurzweil integrals with infinite bounds such as :\int_a^ f(x)\,dx := \lim_\int_a^bf(x)\,dx. For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if is bounded with compact support, the following are equivalent: * is Henstock–Kurzweil integrable, * is Lebesgue integrable, * is
Lebesgue measurable 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 ...
. In general, every Henstock–Kurzweil integrable function is measurable, and is Lebesgue integrable if and only if both and are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a " non-absolutely convergent version of the Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Inform ...
(without requiring the functions to be nonnegative) and
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
(where the condition of dominance is loosened to for some integrable ''g'', ''h''). If ''F'' is differentiable everywhere (or with countably 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: :F(x) - F(a) = \int_a^x F'(t) \,dt. Conversely, the
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. The theorem is named for ...
continues to hold for the Henstock–Kurzweil integral: if ''f'' is Henstock–Kurzweil integrable on , and :F(x) = \int_a^x f(t)\,dt, then ''F''′(''x'') = ''f''(''x'') almost everywhere in (in particular, ''F'' is differentiable almost everywhere). The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm, with respect to which it is barrelled but incomplete.


Utility

The gauge integral has increased utility when compared to the Riemann Integral in that the gauge integral of any function which has a constant value ''c'' except possibly at a countable number of points C = \ can be calculated. Consider for example the piece-wise function f(t) = \begin 0, & \text t \in ,1\text\\ 1, & \text t \in ,1\text \end This function is impossible to integrate using a Riemann Integral because it is impossible to make intervals _, u_i/math> small enough to encapsulate the changing values of ''f''(''x''). With the mapping nature of \delta-fine tagged partitions. The value of the type of integral described above is equal to c(b-a), where ''c'' is the constant value of the function, and ''a, b'' are the function's end points. To demonstrate this, let \varepsilon > 0 be given and let D = \ be a \delta-fine tagged partition of
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> with tags z_j and intervals J_j, and let f(t) be the piecewise function described above. Consider that \left, \sum f(z_j) l(J_j) - 1(1-0)\ = \left, \sum (z_j)-1l(J_j)\ where l(J_j) represents the length of interval J_j. Note this equivalence is established because the summation of the differences in length of all intervals J_j is equal to the length of the interval (or (1-0)). By the definition of the gauge integral, we want to show that the above equation is less than any given \varepsilon. This produces two cases: Case 1: z_j \notin C (All tags of D are irrational): If none of the tags of the tagged partition D are rational, then f(z_j) will always be 1 by the definition of f(t), meaning (z_j)-1= 0. If this term is zero, then for any interval length, the following inequality will be true: \left, \sum (z_j)-1l(J_j)\ \leq \varepsilon, So for this case, 1 is the integral of f(t). Case 2: z_k = c_k (Some tag of D is rational): If a tag of D is rational, then the function evaluated at that point will be 0, which is a problem. Since we know D is \delta-fine, the inequality \left, \sum (z_j)-1l(J_j)\ \leq \left, \sum (z_j)-1l(\delta(c_k))\ holds because the length of any interval J_j is shorter than its covering by the definition of being \delta-fine. If we can construct an \varepsilon out of the right side of the inequality, then we can show the criteria are met for an integral to exist. To do this, let \gamma_k = \varepsilon / (c_k)-c^ and set our covering gauges \delta(c_k) = (c_k-\gamma_k, c_k+\gamma_k), which makes \left, \sum (z_j)-c(J_j)\ < \varepsilon /2^. From this, we have that \left, \sum (z_j)-1l(J_j)\ \leq 2 \sum \varepsilon / 2^ = \varepsilon Because 2 \sum 1 / 2^ = 1 as a
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...
. This indicates that for this case, 1 is the integral of f(t). Since cases 1 and 2 are exhaustive, this shows that the integral of f(t) is 1 and all properties from the above section hold.


McShane integral

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 ...
on a line can also be presented in a similar fashion. If we take the definition of the Henstock–Kurzweil integral from above, and we drop the condition :t_i \in _, u_i then we get a definition of the McShane integral, which is equivalent to the Lebesgue integral. Note that the condition :\forall i \ \ _, u_i\subset _i-\delta(t_i), t_i + \delta (t_i)/math> does still apply, and we technically also require t_i \in ,b/math> for f(t_i) to be defined.


See also

* Pfeffer integral *
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand , ...
*
Hadamard finite part integral In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by . showed that this ca ...


References


Footnotes


General

*
A Modern Integration Theory in 21st Century
* * * * * * * * * * * * *


External links

The following are additional resources on the web for learning more: *
An Introduction to The Gauge Integral

An Open Suggestion: To replace the Riemann integral with the gauge integral in calculus textbooks
signed by Bartle, Henstock, Kurzweil, Schechter, Schwabik, and Výborný {{DEFAULTSORT:Henstock-Kurzweil integral Definitions of mathematical integration