Henstock-Kurzweil integral

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In mathematics, the Henstock-Kurzweil integral, also known as the Denjoy integral (pronounced [dɑ̃ˈʒwa]) and the Perron integral, is a possible definition of the integral of a function. It is a generalisation 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

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 [−ε,δ] and then let ε, δ → 0 (this is called conditional integrability). In effect, the definitions of Denjoy and Lebesgue agree completely on positive functions.

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. 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.

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[edit] Definition

Henstock's definition is as follows:

Given a tagged partition P of [a, b], say

a = u_0 < u_1 < \cdots < u_n = b, \ \ t_i \in [u_{i-1}, u_i]

and a positive function

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

which we call a gauge, we say P is δ-fine if

\forall i \ \ u_i - u_{i-1} < \delta (t_i).

For a tagged partition P and a function

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

we define the Riemann sum to be

\sum_P f = \sum_{i = 1}^n (u_i - u_{i-1}) f(t_i).

Given a function

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

we now define a number I to be the gauge integral of f if for every ε > 0 there exists a gauge δ such that whenever P is δ-fine, we have

{\Big \vert} \sum_P f - I {\Big \vert} < \varepsilon.

The Riemann integral can be regarded as the special case where we only allow constant gauges. Note that due to Cousin's lemma, which says that for every gauge δ there is a δ-fine partition, this condition cannot be satisfied vacuously.

[edit] Properties

Let f \colon [a,b] \to \mathbb{R} be any function.

If f is improperly Riemann integrable, then f is gauge integrable. Moreover, if f is improperly gauge integrable, then f is properly gauge integrable as a consequence of Hake's theorem. This shows that there is no sense in studying an "improper gauge integral".

If ƒ is bounded, then the following are equivalent: (i) ƒ is Lebesgue-integrable, (ii) ƒ is gauge-integrable, (iii) ƒ is measurable.

In general, ƒ is Lebesgue integrable if and only if both ƒ and |ƒ| are gauge integrable.

If ƒ is differentiable everywhere, its derivative ƒ' is gauge-integrable. (Note that ƒ' need not be Lebesgue-integrable.) This leads to 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.

If a < c < b, then f is gauge integrable on [a,b] if and only if it is gauge integrable on both [a,c] and [c,b].

[edit] External links

The following are additional resources on the web for learning more:

[edit] References

  • Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9. 
  • McLeod, Robert M. (1980). The generalized Riemann integral, Carus Mathematical Monographs, 20. Washington, D.C.: Mathematical Association of America. ISBN 0-8838-5021-4. 


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