ThmDex – An index of mathematical definitions, results, and conjectures.
Measurable transformation preserves independence
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $M_j = (\Xi_j, \mathcal{F}_{\Xi_j})$ is a D1108: Measurable space for each $j \in J$
(ii) $X_j : \Omega \to \Xi_j$ is a D202: Random variable from $P$ to $M_j$ for each $j \in J$
(iii) $\{ X_j \}_{j \in J}$ is an D2713: Independent random collection on $P$
(iv) $f_j : \Xi_j \to \Theta_j$ is a D201: Measurable map on $M_j$ for each $j \in J$
Then $\{ f_j(X_j) \}_{j \in J}$ is an D2713: Independent random collection on $P$.