ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4920 on D2713: Independent random collection
Measurable transformation preserves independent finite random collection
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $M_1 = (\Xi_1, \mathcal{F}_{\Xi_1}), \quad \ldots, \quad M_N = (\Xi_N, \mathcal{F}_{\Xi_N})$ are each a D1108: Measurable space
(ii) $X_1 : \Omega \to \Xi_1, \quad \ldots, \quad X_N : \Omega \to \Xi_N$ are each a D202: Random variable from $P$ to $M_1, \ldots, M_N$, respectively
(iii) $X_1, \ldots, X_N$ is an D2713: Independent random collection on $P$
(iv) $f_1 : \Xi_1 \to \Theta_1, \quad \ldots, \quad f_N : \Xi_N \to \Theta_N$ are each a D201: Measurable map on $M_1, \ldots, M_N$, respectively
Then $f_1(X_1), \ldots, f_N(X_N)$ is an D2713: Independent random collection on $P$.
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $M_1 = (\Xi_1, \mathcal{F}_{\Xi_1}), \quad \ldots, \quad M_N = (\Xi_N, \mathcal{F}_{\Xi_N})$ are each a D1108: Measurable space
(ii) $X_1 : \Omega \to \Xi_1, \quad \ldots, \quad X_N : \Omega \to \Xi_N$ are each a D202: Random variable from $P$ to $M_1, \ldots, M_N$, respectively
(iii) $X_1, \ldots, X_N$ is an D2713: Independent random collection on $P$
(iv) $f_1 : \Xi_1 \to \Theta_1, \quad \ldots, \quad f_N : \Xi_N \to \Theta_N$ are each a D201: Measurable map on $M_1, \ldots, M_N$, respectively