Riemann Integration
in One Real Variable

This page is a sub-page of our page on Calculus of One Real Variable.

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Other relevant sources of information:

The fundamental theorem of calculus
• The Riemann integral
• The Riemann-Stiltjes integral
• The Lebesgue integral
• The Lebesgue-Stiltjes integral
Measure
• The gauge integral

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Integration

Riemann-integrable function:

The definite (Riemann) integral of a function:

Converging to the common limit of the upper and lower sums:

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Essence of calculus, chapter 1 (3Blue1Brown on YouTube):

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The gauge integral

Quoting Wikipedia (Henstock–Kurzweil integral):

In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral, 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.

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:

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) = \dfrac{1}{x} \sin \dfrac{1}{x^3} .

This function has a singularity at 0 , and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval \, [−\epsilon,\delta] \, and then let \, \epsilon, \delta \rightarrow 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.

Kurzweil’s definition of the Gauge integral

/////// Quoting Wikipedia

Given a tagged partition \, P \, of \, [a, b] , that is,

\, a = u_0 < u_1 < ⋯ < u_n = b \,

together with \, t_i \in [u_{i−1}, u_i] ,

we define the Riemann sum for a function

\, f : [ a , b] \rightarrow {\mathbb {R}} \,

to be

\, \sum\limits_{P}^{} f = \sum\limits_{i=1}^{n} f(t_i) \Delta u_i ,

where

\, \Delta u_i := u_i − u_{i−1} .

Given a positive function

\, \delta : [ a , b ] \rightarrow ( 0, \infty ) ,

which we call a gauge, we say a tagged partition \, P \, is \, \delta -fine if

\, \forall i \; [ u_{i−1} , u_i ] \subset [ t_i − \delta(t_i) , t_i + \delta(t_i)] .

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

\, | ∑ P f − I | < \varepsilon .

If such an \, I \, exists, we say that \, f \, is Henstock–Kurzweil integrable on \, [a, b] .

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

/////// End of Quote from Wikipedia

The simplicity of Kurzweil's definition has made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.
[An Open letter to the authors of calculus textbooks].

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/////// End of quote from Wikipedia

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