Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space.
An D4361: Unsigned basic function $\mu : \Omega \times \mathcal{F}_{\Xi} \to [0, \infty]$ is a random probability measure on $M$ with respect to $P$ if and only if
| (1) | For every $E \in \mathcal{F}_{\Xi}$, the D4361: Unsigned basic function $\Omega \to [0, \infty]$ given by $\omega \to \mu(\omega, E)$ is a D4381: Random basic number on $P$ |
| (2) | For every $\omega \in \Omega$, the D4361: Unsigned basic function $\mathcal{F}_{\Xi} \to [0, \infty]$ given by $E \mapsto \mu(\omega, E)$ is an D198: Probability measure on $M$ |