Khinchin integral

Definition

Let $a < b$ be real numbers. Let $F \colon [a, b] \to \R$ be a function and let $E \subseteq [a, b]$. The function $F$ is absolutely continuous on $E$ if the following holds. For every $\varepsilon > 0$ there is $\delta > 0$ such that for every finite family of non-overlapping intervals $[a_1, b_1], \dots, [a_n, b_n]$ with all endpoints in $E$, $$\sum_{k=1}^{n} (b_k - a_k) < \delta \quad \text{implies} \quad \sum_{k=1}^{n} \abs{F(b_k) - F(a_k)} < \varepsilon.$$ The function $F$ is generalized absolutely continuous, or ACG, on $[a, b]$ if two conditions hold. First, $F$ is continuous on $[a, b]$. Second, $[a, b]$ is a countable union of sets on each of which $F$ is absolutely continuous. Let $f \colon [a, b] \to \R$ be a function. It is Khinchin integrable if some ACG function $F$ on $[a, b]$ has approximate derivative equal to $f(x)$ at almost every $x \in [a, b]$. The Khinchin integral of $f$ is then $\int_a^b f = F(b) - F(a)$.