Henstock–Kurzweil integral explained

In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced in French pronounced as /dɑ̃ˈʒwa/), 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 of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable over a subset of

\R ^{n} if and only if the function and its absolute value 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 :

$$f(x)\; =\; \backslash frac\backslash sin\backslash left(\backslash frac\backslash right).$$

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 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 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.^[1] 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.^[2]

Definition

Following, given a tagged partition of, that is,together with each subinterval's tag defined as a point$$t\_i\; \backslash in\; [u\_\{i-1\},\; u\_i],$$we define the Riemann sum for a function

f \colon [a, b] \to R to be$$\backslash sum\_P\; f\; =\; \backslash sum\_^n\; f(t\_i)\; \backslash Delta\; u\_i.$$where \Delta u _{i} : = u _{i} - u _{i-1} . This is the summation of each subinterval's length multiplied by the function evaluated at that subinterval's tag .

Given a positive function$$\backslash delta\; \backslash colon\; [a,\; b]\; \backslash to\; (0,\; \backslash infty),$$which we call a gauge, we say a tagged partition P is

\delta -fine if$$(\backslash forall\; i\; \backslash in\; \backslash )\; \backslash \; (\backslash \; [u\_\{i-1\},\; u\_i]\; \backslash subset\; [t\_i-\backslash delta(t\_i),\; t\_i\; +\; \backslash delta\; (t\_i)]).$$

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

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,$$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; \backslash int\_a^c\; f(x)\backslash ,dx\; +\; \backslash int\_c^b\; f(x)\backslash ,dx.$$

Henstock–Kurzweil integrals are linear: given integrable functions, and real numbers,, the expression is integrable ; for example,$$\backslash int\_a^b\; \backslash left(\backslash alpha\; f(x)\; +\; \backslash beta\; g(x)\backslash right)\; dx\; =\; \backslash alpha\; \backslash int\_a^bf(x)\backslash ,dx\; +\; \backslash beta\; \backslash int\_a^b\; g(x)\backslash ,dx.$$

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$$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; \backslash lim\_\; \backslash int\_a^c\; f(x)\backslash ,dx$$

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$$\backslash int\_0^1\; \backslash fracx\backslash ,dx$$

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$$\backslash int\_a^\; f(x)\backslash ,dx\; :=\; \backslash lim\_\backslash int\_a^b\; f(x)\backslash ,dx.$$

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 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 (without requiring the functions to be nonnegative) and dominated convergence theorem (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:$$F(x)-F(a)\; =\; \backslash int\_a^x\; F\text{'}(t)\; \backslash ,dt.$$

Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on, and

$$F(x)\; =\; \backslash int\_a^x\; f(t)\backslash ,dt,$$

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 = \{c _{i:} i \in \N\} can be calculated. Consider for example the piecewise functionwhich is equal to one minus the Dirichlet function on the interval.

This function is impossible to integrate using a Riemann integral because it is impossible to make intervals

[u _{i-1} , u _{i]} 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 = \{(z _{j,} J _{j):} 1 \leq j \leq n\} be a \delta -fine tagged partition of [ 0 , 1 ] with tags z _{j} and intervals J _{j} , and let f(t) be the piecewise function described above. Consider that $$\backslash left|\backslash sum\; f(z\_j)\; l(J\_j)\; -\; 1(1-0)\backslash right|\; =\; \backslash left|\backslash sum\; [f(z\_j)-1]\; l(J\_j)\backslash right|$$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):

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 f(z _{j)-1} = 0 . If this term is zero, then for any interval length, the following inequality will be true:$$\backslash left|\backslash sum\; [f(z\_j)-1]\; l(J\_j)\backslash right|\; \backslash leq\; \backslash varepsilon,$$

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 $$\backslash left|\backslash sum\; [f(z\_j)-1]\; l(J\_j)\backslash right|\; \backslash leq\; \backslash left|\backslash sum\; [f(z\_j)-1]\; l(\backslash delta(c\_k))\backslash right|$$ 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 / [f(c ^{k +2} _{k) - c] 2} and set our covering gauges \delta(c _{k)} = (c _{k} - \gamma _{k,} c _{k} + \gamma _{k)} , which makes

From this, we have that $$\backslash left|\backslash sum\; [f(z\_j)-1]\; l(J\_j)\backslash right|\; \backslash leq\; 2\backslash sum\; \backslash varepsilon\; /\; 2^\; =\; \backslash varepsilon$$

Because $$2\backslash sum\; 1\; /\; 2^\; =\; 1$$ as a geometric series. 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 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

$$t\_i\; \backslash in\; [u\_\{i-1\},\; u\_i],$$

then we get a definition of the McShane integral, which is equivalent to the Lebesgue integral. Note that the condition

$$\backslash forall\; i\; \backslash \; \backslash \; [u\_\{i-1\},\; u\_i]\; \backslash subset\; [t\_i-\backslash delta(t\_i),\; t\_i\; +\; \backslash delta\; (t\_i)]$$

does still apply, and we technically also require $t\_i\; \backslash in\; [a,b]$ for $f(t\_i)$ to be defined.

See also

• Pfeffer integral
• Cauchy principal value
• Hadamard finite part integral

References

General

• Book: Bartle, Robert G. . Robert G. Bartle. A Modern Theory of Integration . . 32 . American Mathematical Society . 2001 . 978-0-8218-0845-0.
• A Modern Integration Theory in 21st Century
• Book: Robert G. . Bartle . Robert G. Bartle . Donald R. . Sherbert . Introduction to Real Analysis . Wiley . 3rd . 1999 . 978-0-471-32148-4 . registration .
• Book: V G . Čelidze . A G . Džvaršeǐšvili. The Theory of the Denjoy Integral and Some Applications . Series in Real Analysis . 3. World Scientific Publishing Company . 1989 . 978-981-02-0021-3.
• Book: A.G. . Das. The Riemann, Lebesgue, and Generalized Riemann Integrals . Narosa Publishers . 2008 . 978-81-7319-933-2 .
• Book: Gordon, Russell A. . The integrals of Lebesgue, Denjoy, Perron, and Henstock . Graduate Studies in Mathematics . 4 . American Mathematical Society . Providence, RI . 1994 . 978-0-8218-3805-1 . registration .
• Book: Henstock, Ralph. Ralph Henstock. Lectures on the Theory of Integration. Series in Real Analysis. 1. World Scientific Publishing Company. 1988. 978-9971-5-0450-2. registration.
• Book: Kurzweil, Jaroslav . Jaroslav Kurzweil . Henstock–Kurzweil Integration: Its Relation to Topological Vector Spaces . Series in Real Analysis . 7. World Scientific Publishing Company . 2000 . 978-981-02-4207-7.
• Book: Kurzweil, Jaroslav . Jaroslav Kurzweil . Integration Between the Lebesgue Integral and the Henstock–Kurzweil Integral: Its Relation to Locally Convex Vector Spaces. Series in Real Analysis . 8. World Scientific Publishing Company . 2002 . 978-981-238-046-3.
• Book: Leader, Solomon . The Kurzweil–Henstock Integral & Its Differentials . Pure and Applied Mathematics Series . CRC . 2001 . 978-0-8247-0535-0 .
• Book: Lee, Peng-Yee . Lanzhou Lectures on Henstock Integration. Series in Real Analysis . 2. World Scientific Publishing Company . 1989 . 978-9971-5-0891-3.
• Book: Lee . Peng-Yee . Výborný. Rudolf . Integral: An Easy Approach after Kurzweil and Henstock . registration . Australian Mathematical Society Lecture Series . Cambridge University Press . 2000. 978-0-521-77968-5.
• Book: McLeod, Robert M. . The generalized Riemann integral . Carus Mathematical Monographs. 20 . Mathematical Association of America . Washington, D.C. . 1980 . 978-0-88385-021-3 .
• Book: Swartz, Charles W. . Introduction to Gauge Integrals . World Scientific Publishing Company . 2001 . 978-981-02-4239-8 .
• Book: Charles W. . Swartz. Douglas S. . Kurtz . Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane. Series in Real Analysis . 9. World Scientific Publishing Company . 2004 . 978-981-256-611-9 .

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

Notes and References

1. Book: Generalized ordinary differential equations in abstract spaces and applications . 2021 . Everaldo M. Bonotto, Marcia Federson, Jacqueline G. Mesquita . 978-1-119-65502-2 . Hoboken, NJ . 1–3. 1269499134.
2. Web site: An Open Letter to Authors of Calculus Books. 27 February 2014.