Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers such that

(i)	$J^+$ is the D3763: Jordan outer measure in $\mathbb{R}^N$
(ii)	$\mathcal{J} = \mathcal{J}(\mathbb{R}^N)$ is the D3765: Set of Jordan measurable sets in $\mathbb{R}^N$

The Jordan measure in $\mathbb{R}^N$ is the D4361: Unsigned basic function \begin{equation} \mathcal{J} \to [0, \infty], \quad E \mapsto J^+(E) \end{equation}