In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ˈʒwa]), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of 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 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

\( f(x)={\frac {1}{x}}\sin \left({\frac {1}{x^{3}}}\right). \)

This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval [−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 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.[1]

Definition

Given a tagged partition P of [a, b], that is,

\( {\displaystyle a=u_{0}<u_{1}<\cdots <u_{n}=b} \)

together with

\( {\displaystyle t_{i}\in [u_{i-1},u_{i}],} \)

we define the Riemann sum for a function

\( f\colon [a,b]\to {\mathbb {R}} \)

to be

\( {\displaystyle \sum _{P}f=\sum _{i=1}^{n}f(t_{i})\Delta u_{i}.} \)

where

\( {\displaystyle \Delta u_{i}:=u_{i}-u_{i-1}.} \)

Given a positive function

\( \delta \colon [a,b]\to (0,\infty ),\, \)

which we call a gauge, we say a tagged partition P is δ {\displaystyle \delta } \delta -fine if

\( \forall i \ \ [u_{i-1}, u_i] \subset [t_i-\delta(t_i), t_i + \delta (t_i)]. \)

We now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge \( \delta \) such that whenever P is \( \delta \) -fine, we have

\( {{\Big \vert }}\sum _{P}f-I{{\Big \vert }}<\varepsilon \) .

If such an I exists, we say that f is Henstock–Kurzweil integrable on [a, b].

Cousin's theorem states that for every gauge δ {\displaystyle \delta } \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 f: [a, b] → ℝ be any function.

Given a < c < b, f is Henstock–Kurzweil integrable on [a, b] if and only if it is Henstock–Kurzweil integrable on both [a, c] and [c, b]; 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 f, g and real numbers α, β, the expression αf + βg is integrable; for example,

\( \int _{a}^{b}\alpha f(x)+\beta g(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx+\beta \int _{a}^{b}g(x)\,dx. \)

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

\( \int _{a}^{b}f(x)\,dx=\lim _{{c\to b^{-}}}\int _{a}^{c}f(x)\,dx \)

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

\( \int _{0}^{1}{\frac {\sin(1/x)}x}\,dx \)

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

\( \int _{a}^{{\infty }}f(x)\,dx:=\lim _{{b\to \infty }}\int _{a}^{b}f(x)\,dx. \)

For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if f is bounded with compact support, the following are equivalent:

f is Henstock–Kurzweil integrable,

f is Lebesgue integrable,

f is Lebesgue measurable.

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 (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 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 continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and

\( F(x)=\int _{a}^{x}f(t)\,dt, \)

then F′(x) = f(x) almost everywhere in [a, b] (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.

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

\( {\displaystyle t_{i}\in [u_{i-1},u_{i}],} \)

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

\( {\displaystyle \forall i\ \ [u_{i-1},u_{i}]\subset [t_{i}-\delta (t_{i}),t_{i}+\delta (t_{i})]} \)

does still apply, and we technically also require \( {\textstyle t_{i}\in [a,b]} \) for \( {\textstyle f(t_{i})} \) to be defined.

See also

Pfeffer integral

Cauchy principal value

Hadamard finite part integral

References

Footnotes

"An Open Letter to Authors of Calculus Books". Retrieved 27 February 2014.

General

Bartle, Robert G. (2001). A Modern Theory of Integration. Graduate Studies in Mathematics. 32. American Mathematical Society. ISBN 978-0-8218-0845-0.

A Modern Integration Theory in 21st Century

Bartle, Robert G.; Sherbert, Donald R. (1999). Introduction to Real Analysis (3rd ed.). Wiley. ISBN 978-0-471-32148-4.

Čelidze, V G; Džvaršeǐšvili, A G (1989). The Theory of the Denjoy Integral and Some Applications. Series in Real Analysis. 3. World Scientific Publishing Company. ISBN 978-981-02-0021-3.

Das, A.G. (2008). The Riemann, Lebesgue, and Generalized Riemann Integrals. Narosa Publishers. ISBN 978-81-7319-933-2.

Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics. 4. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3805-1.

Henstock, Ralph (1988). Lectures on the Theory of Integration. Series in Real Analysis. 1. World Scientific Publishing Company. ISBN 978-9971-5-0450-2.

Kurzweil, Jaroslav (2000). Henstock–Kurzweil Integration: Its Relation to Topological Vector Spaces. Series in Real Analysis. 7. World Scientific Publishing Company. ISBN 978-981-02-4207-7.

Kurzweil, Jaroslav (2002). 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. ISBN 978-981-238-046-3.

Leader, Solomon (2001). The Kurzweil–Henstock Integral & Its Differentials. Pure and Applied Mathematics Series. CRC. ISBN 978-0-8247-0535-0.

Lee, Peng-Yee (1989). Lanzhou Lectures on Henstock Integration. Series in Real Analysis. 2. World Scientific Publishing Company. ISBN 978-9971-5-0891-3.

Lee, Peng-Yee; Výborný, Rudolf (2000). Integral: An Easy Approach after Kurzweil and Henstock. Australian Mathematical Society Lecture Series. Cambridge University Press. ISBN 978-0-521-77968-5.

McLeod, Robert M. (1980). The generalized Riemann integral. Carus Mathematical Monographs. 20. Washington, D.C.: Mathematical Association of America. ISBN 978-0-88385-021-3.

Swartz, Charles W. (2001). Introduction to Gauge Integrals. World Scientific Publishing Company. ISBN 978-981-02-4239-8.

Swartz, Charles W.; Kurtz, Douglas S. (2004). Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane. Series in Real Analysis. 9. World Scientific Publishing Company. ISBN 978-981-256-611-9.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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