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}