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 $A$.

Remarks

Here $\sup$ is the supremum and $\inf$ is the infimum. Closed boxes, balls and bounded regions between the graphs of two continuous functions are Jordan measurable. Finite unions, intersections and differences of Jordan measurable sets are Jordan measurable. A bounded set $A$ is Jordan measurable if and only if its boundary has Jordan measure zero. Then $m(A) = \int_Q \mathbf{1}_A$ for each closed box $Q \supseteq A$, where $\mathbf{1}_A$ is the indicator function of $A$. The set $\mathbb{Q} \cap [0, 1]$ is not Jordan measurable: its inner measure is $0$ and its outer measure is $1$. Lebesgue measure extends Jordan measure to more sets.

Explanations

L1 human

Jordan measure measures a set by fitting finite unions of boxes inside it and around it. If the best inner and outer approximations have the same total volume, that volume is the measure of the set.