---
tag: Q1TV4F608D
kind: concept
name: Khinchin integral
msc: [26A39]
sources:
  - url: https://en.wikipedia.org/wiki/Khinchin_integral
same_as:
  wikidata: Q6401692
  wikipedia: Khinchin integral
---

## Summary

The Khinchin integral is an integral on an interval defined through primitives rather than through sums.
It is also called the Denjoy–Khinchin integral or the wide Denjoy integral.
A function is Khinchin integrable when it is, almost everywhere, the [[approximate derivative|4W31MKE01V]] of a continuous function that is generalized [[absolutely continuous|ZT6F4C2YHK]].
It extends the [[Henstock–Kurzweil integral|FCGHXDGVV9]], and hence the [[Lebesgue integral|DWJF7ABPR1]] and the [[Perron integral|M077AXJACA]].

## Explanations

### L1 human

In Lebesgue's theory, the indefinite integrals are exactly the absolutely continuous functions, and each one has the integrand as its derivative almost everywhere.
The Khinchin integral relaxes this in two ways.
The primitive only has to be absolutely continuous on each of countably many pieces of the interval.
It only has to have the function as an approximate derivative, a derivative that may ignore thin sets of points.
The integral of $f$ is then the change of the primitive from one end of the interval to the other.