User:Caliburn/s/prob/Definition:Independent Sigma-Algebras/Binary Case
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG_1$ and $\GG_2$ be sub-$\sigma$-algebras of $\EE$.
Then $\GG_1$ and $\GG_2$ are said to be ($\Pr$-)independent if and only if:
- $\forall E_1 \in \GG_1, E_2 \in \GG_2: \map \Pr {E_1 \cap E_2} = \map \Pr {E_1} \map \Pr {E_2}$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 5$: Problem $10$