Peano-Jordan Measure#

Theorem: Jordan-Measurability via Riemann Integrals

A subset \(S \subseteq \mathbb{R}^n\) is Jordan-measurable if and only if there exists a compact rectangular region \(R \subset \mathbb{R}^n\) containing \(S\) such that the indicator function \(\mathbf{1}_S : \mathbb{R}^n \to \mathbb{R}\)

is Riemann-integrable on \(R\). In this case, the value of this integral is the Peano-Jordan measure of \(S\).