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 map
Random variable
Random number
Random Euclidean number
Random basic number
Definition D3161
Random real number
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be the D5072: Standard real borel measurable space.
A D4364: Real function $X : \Omega \to \mathbb{R}$ is a random real number on $P$ if and only if \begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}) : X^{-1}(E) \in \mathcal{F} \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be the D5072: Standard real borel measurable space.
A D4364: Real function $X : \Omega \to \mathbb{R}$ is a random real number on $P$ if and only if \begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}) : \{ X \in E \} \in \mathcal{F} \end{equation}
Formulation 2
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be the D5072: Standard real borel measurable space.
A D4364: Real function $X : \Omega \to \mathbb{R}$ is a random real number on $P$ if and only if \begin{equation} \sigma_{\text{pullback}, M} \langle X \rangle \subseteq \mathcal{F} \end{equation}
Children
Random rational number
Results
R4754
Interval length upper bound to variance of bounded random real number