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}

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}

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}