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

${\displaystyle f(x)=\int _{a}^{x}g(t)\,dt}$

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

1. f is absolutely continuous (see below)
2. f is differentiable almost everywhere
3. 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)=x²·sin(1/x²) (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

$\sum _{k}\left|y_{k}-x_{k}\right|<\delta$

it also satisfies

${\displaystyle \sum _{k}|f(y_{k})-f(x_{k})|<\varepsilon .}$

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

${\displaystyle \lim _{\varepsilon \to 0}{\frac {\mu (E\cap [x-\varepsilon ,x+\varepsilon ])}{2\varepsilon }}=1}$

(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 ${\displaystyle g^{-1}([y-\varepsilon ,y+\varepsilon ])}$. (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

${\displaystyle {\frac {f(x')-f(x)}{x'-x}}}$

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

1. (Gordon 1994, theorem 4.12)
2. (Gordon 1994, theorem 4.14)
3. (Bruckner 1994, chapter 5, §2)
4. (Bruckner 1994, chapter 5, §4)
5. (Gordon 1994, definition 6.1)
6. (Gordon 1994, theorem 6.10)
7. A portion of a perfect set P is a P ∩ [u, v] such that this intersection is perfect and nonempty.
8. (Bruckner 1994, chapter 10, §1)
9. (Gordon 1994, theorem 14.5)
10. (Bruckner 1994, theorem 5.2)
11. (Gordon 1994, theorem 14.7)
12. (Bruckner 1994, chapter 10, theorem 1.2)
13. (Gordon 1994, theorem 14.11)
14. (Filippov 1998, chapter IV, theorem 6.1)
15. (Gordon 1994, theorem 15.2)
16. (Gordon 1994, definition 15.1)
17. (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.