Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space.
A D1721: Random collection $X : J \to \text{Random}(P \to M)$ is identically distributed if and only if \begin{equation} \forall \, i, j \in J : \forall \, E \in \mathcal{F}_{\Xi} : \mathbb{P}(X_i \in E) = \mathbb{P}(X_j \in E) \end{equation}