Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
| (i) | $M_n = (\Xi_n, \mathcal{F}_{\Xi_n})$ is a D1108: Measurable space for each $n \in \mathbb{N}$ |
| (ii) | $X_n : \Omega \to \Xi_n$ is a D202: Random variable from $P$ to $M_n$ for each $n \in \mathbb{N}$ |
| (iii) | $\{ X_n \}_{n \in \mathbb{N}}$ is an D2713: Independent random collection on $P$ |
| (iv) | $f_n : \Xi_n \to \Theta_n$ is a D201: Measurable map on $M_n$ for each $n \in \mathbb{N}$ |
Then $\{ f_n (X_n) \}_{n \in \mathbb{N}}$ is an D2713: Independent random collection on $P$.