Jordan measure

Definition

A box in $\mathbb{R}^n$ is a set $Q = [a_1, b_1) \times \dots \times [a_n, b_n)$ with $a_i \le b_i$, and its volume is $|Q| = (b_1 - a_1) \cdots (b_n - a_n)$. A simple set is a finite union of boxes. Each simple set $S$ is a finite union of pairwise disjoint boxes $Q_1, \dots, Q_k$, and $m(S) = |Q_1| + \dots + |Q_k|$ does not depend on this choice. For a bounded set $A \subseteq \mathbb{R}^n$, the inner and outer Jordan measures are $$m_*(A) = \sup \{ m(S) : S \subseteq A \}, \qquad m^*(A) = \inf \{ m(S) : S \supseteq A \},$$ where $S$ runs over simple sets. The set $A$ is Jordan measurable if $m_*(A) = m^*(A)$, and then this common value $m(A)$ is the Jordan measure, or Jordan content, of $A$.