Henstock-Kurzweil integral

From Academic Kids

In mathematics, the Henstock-Kurzweil integral, also known as the Denjoy integral (pronounce Denjua) 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

<math>\frac{1}{x}\sin\left(\frac{1}{x^3}\right).<math>

This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over <math>[-\epsilon,\epsilon]<math> and then let ε → 0 (this is called principal value integration or 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 never quite popularized.

Another important property of the Henstock integral is that every function which is the derivative of some other function is gauge integrable, so a very strong form of the fundamental theorem of calculus holds. In particular a non-trivial corollary applies to the Lebesgue integral: if a function f is differentiable everywhere and its derivative is Lebesgue integrable, then f is the integral of its derivative.

Definition

Henstock's definition is as follows:

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

<math>a = u_0 < u_1 < \ldots < u_n = b, \ \ v_i \in [u_{i-1}, u_i]<math>

and a positive function

<math>\delta : [a, b] \to (0, \infty)<math>,

which we call a gauge, we say P is <math>\delta<math>-fine if

<math>\forall i \ \ u_i - u_{i-1} < \delta (v_i) <math>.

For a tagged partition P and a function

<math>f : [a, b] \to \mathbb{R}<math>

we define the Riemann sum to be

<math> \sum_P f = \sum_{i = 1}^n (u_i - u_{i-1}) f(v_i)<math>

Given a function

<math>f : [a, b] \to \mathbb{R}<math>

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

<math>\left| \sum_P f - I \right| < \epsilon. <math>

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 <math>\delta<math> there is a <math>\delta<math>-fine partition, this condition cannot be satisfied vacuously.

External links

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

References

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