Khinchin integral

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 ${\displaystyle [x_{k},y_{k}]}$ of pairwise disjoint subintervals of I with endpoints in E satisfies

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

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.

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.

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.

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 ${\displaystyle f(b)-f(a)}$.

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.