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.

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

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

is its indefinite Lebesgue 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) = 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.

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

\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

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

{\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.

[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)

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.

[1] [1]
(Gordon 1994, theorem 4.12)

[2] [2]
(Gordon 1994, theorem 4.14)

[3] [3]
(Bruckner 1994, chapter 5, §2)

[4] [4]
(Bruckner 1994, chapter 5, §4)

[5] [5]
(Gordon 1994, definition 6.1)

[6] [6]
(Gordon 1994, theorem 6.10)

[7] [7]
A portion of a perfect set P is a P ∩ [u, v] such that this intersection is perfect and nonempty.

[8] [8]
(Bruckner 1994, chapter 10, §1)

[9] [9]
(Gordon 1994, theorem 14.5)

[10] [10]
(Bruckner 1994, theorem 5.2)

[11] [11]
(Gordon 1994, theorem 14.7)

[12] [12]
(Bruckner 1994, chapter 10, theorem 1.2)

[13] [13]
(Gordon 1994, theorem 14.11)

[14] [14]
(Filippov 1998, chapter IV, theorem 6.1)

[15] [15]
(Gordon 1994, theorem 15.2)

[16] [16]
(Gordon 1994, definition 15.1)

[17] [17]
(Gordon 1994, theorem 15.4)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

Khinchin integral