▾ Set of symbols
▾ Alphabet
▾ Deduction system
▾ Theory
▾ Zermelo-Fraenkel set theory
▾ Set
▾ Binary cartesian set product
▾ Binary relation
▾ Map
▾ Function
▾ Measure
▾ Real measure
▾ Euclidean real measure
▾ Complex measure
▾ Basic measure
▾ Unsigned basic measure
Submeasure
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $X$
An D4361: Unsigned basic function $\nu : \mathcal{G} \to [0, \infty]$ is a submeasure of $\mu$ on $M$ with respect to $\mathcal{G}$ if and only if \begin{equation} \forall \, E \in \mathcal{G} : \nu(E) = \mu(E) \end{equation}