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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers.\nLet $F \\colon [a, b] \\to \\R$ be a function and let $E \\subseteq [a, b]$.\nThe function $F$ is *absolutely continuous on $E$* if the following holds.\nFor every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that for every finite family of non-overlapping intervals $[a_1, b_1], \\dots, [a_n, b_n]$ with all endpoints in $E$,\n$$\\sum_{k=1}^{n} (b_k - a_k) \u003c \\delta \\quad \\text{implies} \\quad \\sum_{k=1}^{n} \\abs{F(b_k) - F(a_k)} \u003c \\varepsilon.$$\nThe function $F$ is *generalized absolutely continuous*, or ACG, on $[a, b]$ if two conditions hold.\nFirst, $F$ is [[continuous|0P27W9RW0G]] on $[a, b]$.\nSecond, $[a, b]$ is a countable union of sets on each of which $F$ is absolutely continuous.\nLet $f \\colon [a, b] \\to \\R$ be a function.\nIt is *Khinchin integrable* if some ACG function $F$ on $[a, b]$ has [[approximate derivative|4W31MKE01V]] equal to $f(x)$ at [[almost every|RARTMA0ZEC]] $x \\in [a, b]$.\nThe *Khinchin integral* of $f$ is then $\\int_a^b f = F(b) - F(a)$.\n\n## Remarks\n\nThe integral is well defined: two ACG functions with the same approximate derivative almost everywhere differ by a constant.\n\nExample: let $f$ be Lebesgue integrable on $[a, b]$.\nThen $F(x) = \\int_a^x f$ is [[absolutely continuous|ZT6F4C2YHK]] on all of $[a, b]$, hence ACG, and $F' = f$ almost everywhere.\nSo the Khinchin and Lebesgue integrals of $f$ agree.\n\nNon-example of a primitive: the [[Cantor function|NPBYYK63FE]] $c$ is continuous and $c'(x) = 0$ for almost every $x$, but $c$ is not ACG.\nIf it were, the uniqueness above would force $c$ to be constant, while $c(1) - c(0) = 1$.\n",
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