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
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.
Given a tagged partition P of [a, b], that is,
together with each partition's tag defined as a point
we define the Riemann sum for a function to be
where This is the summation of each partition's length () multiplied by the function evaluated at that partition'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 [a, b].
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.
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,
Henstock–Kurzweil integrals are linear. Given integrable functions f, g and real numbers α, β, the expression αf + βg is integrable; 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 states that
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
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:
- 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:
Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and
then F′(x) = f(x) almost everywhere in [a, b] (in particular, F is differentiable almost everywhere).
The gauge integral has increased utility when compared to the Riemann Integral in that the gauge integral of any function f: [a, b] → R which has a constant value c except possibly at a countable number of points can be calculated. Consider for example the piece-wise function
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 end points. 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 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 an 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
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.
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.
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The following are additional resources on the web for learning more:
- "Kurzweil-Henstock integral", Encyclopedia of Mathematics, EMS Press, 2001 
- 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