ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Interval
Set of intervals
J-interval
N-interval
Real N-interval
Set of euclidean real intervals
Elementary euclidean real set
Elementary euclidean real partition
Elementary euclidean real volume
Elementary euclidean real volume function
Elementary measure
Definition D3763
Jordan outer measure
Formulation 0
Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers such that
(i) $\mathcal{P}_{\text{bounded}}(\mathbb{R}^N)$ is the D3756: Set of bounded euclidean real sets in $\mathbb{R}^N$
(ii) $\mathcal{P}_{\text{elementary}}(\mathbb{R}^N)$ is the D3761: Set of elementary sets in $\mathbb{R}^N$
(iii) $m$ is the D3760: Elementary measure in $\mathbb{R}^N$
The Jordan outer measure in $\mathbb{R}^N$ is the D4361: Unsigned basic function \begin{equation} \mathcal{P}_{\text{bounded}}(\mathbb{R}^N) \to [0, \infty], \quad B \mapsto \inf_{E \in \mathcal{P}_{\text{elementary}}(\mathbb{R}^N) : B \subseteq E} m(E) \end{equation}
Children
D3765: Set of Jordan measurable sets