ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Subset structure
Measurable space
Measurable set
Null measurable set
Subnull set
Definition D1704
Complete measure
Formulation 0
A D85: Unsigned basic measure $\mu : \mathcal{F} \to [0, \infty]$ is complete if and only if \begin{equation} \forall \, F \in \mathcal{F} \left( \mu(F) = 0 \quad \implies \quad \forall \, E \subseteq F : E \in \mathcal{F} \right) \end{equation}
Formulation 1
Let $M = (X, \mathcal{F}, \mu)$ be D1158: Measure space such that
(i) $\mathsf{Subnull} = \mathsf{Subnull}(M)$ is the D3804: Set of subnull sets in $M$
Then $\mu$ is a complete measure if and only if \begin{equation} \mathsf{Subnull} \subseteq \mathcal{F} \end{equation}
Children
D1678: Complete measure space
D3774: Complete probability measure