If g : I → R is a Lebesgue-integrable function on some interval I = [a,b], and if is its indefinite Lebesgue integral, then the following assertions are true:[1] 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.
To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function.
of pairwise disjoint subintervals of I with endpoints in E satisfies it also satisfies 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.
Recall that a real number x (not necessarily in E) is said to be a point of density of E when (where μ denotes Lebesgue measure).
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.
[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.