Denjoy was interested in a definition that would allow one to integrate functions like:
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.
) multiplied by the function evaluated at that subinterval's tag (
We now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge
-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.
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.
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 example, if f is bounded with compact support, the following are equivalent: In general, every Henstock–Kurzweil integrable function is measurable, and
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).
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
The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm, with respect to which it is barrelled but incomplete.
which has a constant value c except possibly at a countable number of points
which is equal to one minus the Dirichlet function on the interval.
small enough to encapsulate the changing values of f(x) with the mapping nature of
Note this equivalence is established because the summation of the consecutive differences in length of all intervals
By the definition of the gauge integral, we want to show that the above equation is less than any given
If this term is zero, then for any interval length, the following inequality will be true:
is rational, then the function evaluated at that point will be 0, which is a problem.
out of the right side of the inequality, then we can show the criteria are met for an integral to exist.
Lebesgue integral on a line can also be presented in a similar fashion.