Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$M = (\Xi, \mathcal{F}_{\Xi})$ is a D1108: Measurable space
(ii)	$X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$

An D4361: Unsigned basic function $\mu : \mathcal{F}_{\Xi} \to [0, \infty]$ is a probability distribution measure for $X$ with respect to $P$ and $M$ if and only if $$\begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X^{-1}(E) ) \end{equation}$$

Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$M = (\Xi, \mathcal{F}_{\Xi})$ is a D1108: Measurable space
(ii)	$X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$

An D4361: Unsigned basic function $\mu : \mathcal{F}_{\Xi} \to [0, \infty]$ is a probability distribution measure for $X$ with respect to $P$ and $M$ if and only if $$\begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X \in E ) \end{equation}$$