Henstock–Kurzweil integral

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In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ʒwa]), 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 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:

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

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Following Bartle (2001), given a tagged partition   of  , that is,  together with each subinterval's tag defined as a point   we define the Riemann sum for a function   to be   where   This is the summation of each subinterval's length ( ) multiplied by the function evaluated at that subinterval's tag ( ).

Given a positive function   which we call a gauge, we say a tagged partition P is  -fine if  

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

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

Cousin's theorem states that for every gauge  , such a  -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

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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 (Bartle 2001, 3.7), 

Henstock–Kurzweil integrals are linear: given integrable functions   and   and real numbers   and  , the expression   is integrable (Bartle 2001, 3.1); for example, 

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 (Bartle 2001, 12.8) states that 

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 

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  

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:

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 g(x) ≤ fn(x) ≤ h(x) for some integrable g, h).

If   is differentiable everywhere (or with countably many exceptions), the derivative   is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is  . (Note that   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: 

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

 

then   almost everywhere in   (in particular,   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

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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   can be calculated. Consider for example the piecewise function  which 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   small enough to encapsulate the changing values of f(x) with the mapping nature of  -fine tagged partitions.

The value of the type of integral described above is equal to  , where c is the constant value of the function, and a, b are the function's endpoints. To demonstrate this, let   be given and let   be a  -fine tagged partition of   with tags   and intervals  , and let   be the piecewise function described above. Consider that   where   represents the length of interval  . Note this equivalence is established because the summation of the consecutive differences in length of all intervals   is equal to the length of the interval (or  ).

By the definition of the gauge integral, we want to show that the above equation is less than any given  . This produces two cases:

Case 1:   (All tags of   are irrational):

If none of the tags of the tagged partition   are rational, then   will always be 1 by the definition of  , meaning  . If this term is zero, then for any interval length, the following inequality will be true:  

So for this case, 1 is the integral of  .

Case 2:   (Some tag of   is rational):

If a tag of   is rational, then the function evaluated at that point will be 0, which is a problem. Since we know   is  -fine, the inequality   holds because the length of any interval   is shorter than its covering by the definition of being  -fine. If we can construct a gauge   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   and set our covering gauges  , which makes  

From this, we have that  

Because   as a geometric series. This indicates that for this case, 1 is the integral of  .

Since cases 1 and 2 are exhaustive, this shows that the integral of   is 1 and all properties from the above section hold.

McShane integral

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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

 

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

 

does still apply, and we technically also require   for   to be defined.

See also

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References

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Footnotes

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  1. ^ Generalized ordinary differential equations in abstract spaces and applications. Everaldo M. Bonotto, Marcia Federson, Jacqueline G. Mesquita. Hoboken, NJ. 2021. pp. 1–3. ISBN 978-1-119-65502-2. OCLC 1269499134.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
  2. ^ "An Open Letter to Authors of Calculus Books". Retrieved 27 February 2014.

General

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