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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (), 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, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
of a function. 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 Gö ...
, and in some situations is more general than the
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
. In particular, a function is Lebesgue integrable over a subset of \R^n
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
are Henstock–Kurzweil integrable. This integral was first defined by Arnaud Denjoy (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 \varepsilon, \delta/math> and then let \varepsilon, \delta \rightarrow 0. 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 Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ...
(using variations on the notions of
absolute continuity In calculus and real analysis, 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 ...
), and by
Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) 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. He made numerous contributions to differentia ...
, 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 Kurzweil named the gauge integral. In 1961 Ralph Henstock independently introduced a similar integral that extended the theory, 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 Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
courses.


Definition

Following , given a tagged partition \mathcal of , b/math>, that is,a = u_0 < u_1 < \cdots < u_n = b together with each subinterval'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 subinterval's length multiplied by the function evaluated at that subinterval's tag . 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 \in \) \ (\ _, 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 , b/math>. Cousin's theorem states that for every gauge \delta, such a \delta-fine partition ''P'' does exist, so this condition cannot be satisfied
vacuously In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a s ...
. The Riemann integral can be regarded as the special case where we only allow constant gauges.


Properties

Let f: , b\to \mathbb be any function. Given a < c < b, f is Henstock–Kurzweil integrable on , b/math> if and only if it is Henstock–Kurzweil integrable on both , c/math> and , b/math>; 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 In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
: given integrable functions f and g and
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s \alpha and \beta, the expression \alpha f + \beta g 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 f 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^b f(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 In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
, 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 higher dimensional Euclidean '-spaces. For lower dimensions or , it coin ...
. 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 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 good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its ...
(without requiring the functions to be nonnegative) and
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
(where the condition of dominance is loosened to for some integrable ''g'', ''h''). If F is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
everywhere (or with countably many exceptions), the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
a constant, the integral of its derivative:F(x)-F(a) = \int_a^x F'(t) \,dt. Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on , b/math>, and F(x) = \int_a^x f(t)\,dt, then F'(x) = f(x)
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
in , b/math> (in particular, F is differentiable almost everywhere). The
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
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 f: , b\mapsto \mathbb which has a constant value ''c'' except possibly at a countable number of points C = \ can be calculated. Consider for example the piecewise functionf(t) = \begin 0, & \text t \in ,1\text\\ 1, & \text t \in ,1\text \end which is equal to one minus the
Dirichlet function In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathb ...
on the interval. 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 endpoints. 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. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/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 consecutive 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 Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
): If none of the tags of the tagged partition D are
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
, then f(z_j) will always be 1 by the definition of f(t), meaning f(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 a gauge \delta 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 a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
. 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 (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
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 * Hadamard finite part integral


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