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Definition D204
Probability distribution measure

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 $$\forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X^{-1}(E) )$$

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 $$\forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X \in E )$$