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
▼ Complete measure
Definition D3774
Complete probability measure
Formulation 0
A D198: Probability measure $\mathbb{P} : \mathcal{F} \to [0, 1]$ is complete if and only if \begin{equation} \forall \, F \in \mathcal{F} \left( \mathbb{P}(F) = 0 \quad \implies \quad \forall \, E \subseteq F : E \in \mathcal{F} \right) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be D1159: Probability space such that
(i) $\mathsf{Subnull} = \mathsf{Subnull}(P)$ is the D3804: Set of subnull sets in $P$
Then $\mathbb{P}$ is a complete probability measure if and only if \begin{equation} \mathsf{Subnull} \subseteq \mathcal{F} \end{equation}