(i) | $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $X$ |

**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}

▾ 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

▾ 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

Formulation 0

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

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}

(i) | $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $X$ |