Khinchin integral

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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[edit]

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 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[edit]

Generalized absolutely continuous function[edit]

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 [xk,yk] 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 Ei such that f is absolutely continuous on each Ei. 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[edit]

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 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

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

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

A theorem[edit]

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[edit]

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[edit]

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[edit]

  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 ∩ [uv] 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[edit]

  • Filippov, V.V. (1998). Basic Topological Structures of Ordinary Differential Equations. ISBN 978-0-7923-4951-8.