Perron integral

Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function. For $U \colon [a, b] \to \R$ and $x \in [a, b]$, the lower and upper derivatives of $U$ at $x$ are $$\underline{D} U(x) = \liminf_{y \to x} \frac{U(y) - U(x)}{y - x}, \qquad \overline{D} U(x) = \limsup_{y \to x} \frac{U(y) - U(x)}{y - x},$$ where $y$ ranges over $[a, b] \setminus \set{x}$. A major function of $f$ is a function $U$ with $\underline{D} U(x) \ne -\infty$ and $\underline{D} U(x) \ge f(x)$ for every $x \in [a, b]$. A minor function of $f$ is a function $V$ with $\overline{D} V(x) \ne +\infty$ and $\overline{D} V(x) \le f(x)$ for every $x \in [a, b]$. Let $\mathcal{U}$ be the set of major functions and $\mathcal{V}$ the set of minor functions of $f$. The function $f$ is Perron integrable on $[a, b]$ if both sets are nonempty and $$\inf_{U \in \mathcal{U}} \bigl(U(b) - U(a)\bigr) = \sup_{V \in \mathcal{V}} \bigl(V(b) - V(a)\bigr) \in \R.$$ The Perron integral of $f$ is this value.