# Khinchin integral

In mathematics, the **Khinchin integral** (sometimes spelled **Khintchine integral**), also known as the **Denjoy–Khinchin integral**, **generalized Denjoy integral** or **wide Denjoy integral**, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann and Lebesgue integrals. It is named after Aleksandr Khinchin and Arnaud Denjoy, but is not to be confused with the (narrow) Denjoy integral.

## Motivation

If *g* : *I* → **R** is a Lebesgue-integrable function on some interval *I* = [*a*,*b*], and if

is its Lebesgue indefinite integral, then the following assertions are true:[1]

*f*is absolutely continuous (see below)*f*is differentiable almost everywhere- Its derivative coincides almost everywhere with
*g*(*x*). (In fact,*all*absolutely continuous functions are obtained in this manner.[2])

The Lebesgue integral could be defined as follows: *g* is Lebesgue-integrable on *I* iff there exists a function *f* that is absolutely continuous whose derivative coincides with *g* almost everywhere.

However, even if *f* : *I* → **R** is differentiable *everywhere*, and *g* is its derivative, it does not follow that *f* is (up to a constant) the Lebesgue indefinite integral of *g*, simply because *g* can fail to be Lebesgue-integrable, i.e., *f* can fail to be absolutely continuous. An example of this is given[3] by the derivative *g* of the (differentiable but not absolutely continuous) function *f*(*x*)=*x*²·sin(1/*x*²) (the function *g* is not Lebesgue-integrable around 0).

The Denjoy integral corrects this lack by ensuring that the derivative of any function *f* that is everywhere differentiable (or even differentiable everywhere except for at most countably many points) is integrable, and its integral reconstructs *f* up to a constant; the Khinchin integral is even more general in that it can integrate the *approximate* derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function. This is the concept of *generalized* absolute continuity; generalized absolutely continuous functions will be exactly those functions which are indefinite Khinchin integrals.

## Definition

### Generalized absolutely continuous function

Let *I* = [*a*,*b*] be an interval and *f* : *I* → **R** be a real-valued function on *I*.

Recall that *f* is absolutely continuous on a subset *E* of *I* if and only if for every positive number *ε* there is a positive number *δ* such that whenever a finite collection [*x*_{k},*y*_{k}] of pairwise disjoint subintervals of *I* with endpoints in *E* satisfies

it also satisfies

Define[4][5] the function *f* to be *generalized absolutely continuous* on a subset *E* of *I* if the restriction of *f* to *E* is continuous (on *E*) and *E* can be written as a countable union of subsets *E*_{i} such that *f* is absolutely continuous on each *E*_{i}. This is equivalent[6] to the statement that every nonempty perfect subset of *E* contains a portion[7] on which *f* is absolutely continuous.

### Approximate derivative

Let *E* be a Lebesgue measurable set of reals. Recall that a real number *x* (not necessarily in *E*) is said to be a *point of density* of *E* when

(where *μ* denotes Lebesgue measure). A Lebesgue-measurable function *g* : *E* → **R** is said to have *approximate limit*[8] *y* at *x* (a point of density of *E*) if for every positive number *ε*, the point *x* is a point of density of . (If furthermore *g*(*x*) = *y*, we can say that *g* is *approximately continuous* at *x*.[9]) Equivalently, *g* has approximate limit *y* at *x* if and only if there exists a measurable subset *F* of *E* such that *x* is a point of density of *F* and the (usual) limit at *x* of the restriction of *f* to *F* is *y*. Just like the usual limit, the approximate limit is unique if it exists.

Finally, a Lebesgue-measurable function *f* : *E* → **R** is said to have *approximate derivative* *y* at *x* iff

has approximate limit *y* at *x*; this implies that *f* is approximately continuous at *x*.

### A theorem

Recall that it follows from Lusin's theorem that a Lebesgue-measurable function is approximately continuous almost everywhere (and conversely).[10][11] The key theorem in constructing the Khinchin integral is this: a function *f* that is generalized absolutely continuous (or even of "generalized bounded variation", a weaker notion) has an approximate derivative almost everywhere.[12][13][14] Furthermore, if *f* is generalized absolutely continuous and its approximate derivative is nonnegative almost everywhere, then *f* is nondecreasing,[15] and consequently, if this approximate derivative is zero almost everywhere, then *f* is constant.

### The Khinchin integral

Let *I* = [*a*,*b*] be an interval and *g* : *I* → **R** be a real-valued function on *I*. The function *g* is said to be Khinchin-integrable on *I* iff there exists a function *f* that is generalized absolutely continuous whose approximate derivative coincides with *g* almost everywhere;[16] in this case, the function *f* is determined by *g* up to a constant, and the Khinchin-integral of *g* from *a* to *b* is defined as *f*(*b*) − *f*(*a*).

### A particular case

If *f* : *I* → **R** is continuous and has an approximate derivative everywhere on *I* except for at most countably many points, then *f* is, in fact, generalized absolutely continuous, so it is the (indefinite) Khinchin-integral of its approximate derivative.[17]

This result does not hold if the set of points where *f* is not assumed to have an approximate derivative is merely of Lebesgue measure zero, as the Cantor function shows.

## Notes

- (Gordon 1994, theorem 4.12)
- (Gordon 1994, theorem 4.14)
- (Bruckner 1994, chapter 5, §2)
- (Bruckner 1994, chapter 5, §4)
- (Gordon 1994, definition 6.1)
- (Gordon 1994, theorem 6.10)
- A
*portion*of a perfect set*P*is a*P*∩ [*u*,*v*] such that this intersection is perfect and nonempty. - (Bruckner 1994, chapter 10, §1)
- (Gordon 1994, theorem 14.5)
- (Bruckner 1994, theorem 5.2)
- (Gordon 1994, theorem 14.7)
- (Bruckner 1994, chapter 10, theorem 1.2)
- (Gordon 1994, theorem 14.11)
- (Filippov 1998, chapter IV, theorem 6.1)
- (Gordon 1994, theorem 15.2)
- (Gordon 1994, definition 15.1)
- (Gordon 1994, theorem 15.4)

## References

- Springer Encyclopedia of Mathematics: article "Denjoy integral"
- Springer Encyclopedia of Mathematics: article "Approximate derivative"
- Bruckner, Andrew (1994).
*Differentiation of Real Functions*. American Mathematical Society. ISBN 978-0-8218-6990-1. - Gordon, Russell A. (1994).
*The Integrals of Lebesgue, Denjoy, Perron, and Henstock*. American Mathematical Society. ISBN 978-0-8218-3805-1. - Filippov, V.V. (1998).
*Basic Topological Structures of Ordinary Differential Equations*. ISBN 978-0-7923-4951-8.