Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
| (i) | $X \in \mathsf{Random}(\mathbb{R}^D)$ is a D4383: Random euclidean real number |
| (ii) | \begin{equation} \forall \, B \in \mathcal{B}(\mathbb{R}^D) \left( \ell(B) = 0 \quad \implies \quad \mathbb{P}(X \in B) = 0 \right) \end{equation} |
Then there exists an D4361: Unsigned basic function $f_X : \mathbb{R}^D \to [0, \infty]$ such that
| (1) | $f_X$ is an D5610: Unsigned basic Borel function on $M$ |
| (2) | \begin{equation} \forall \, B \in \mathcal{B}(\mathbb{R}^D) : \mathbb{P}(X \in B) = \int_B f_X \, d \ell \end{equation} |