User:Caliburn/s/prob/Random Variables Independent iff Independent on Generator Closed under Intersection

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG$ and $\HH$ be sub-$\sigma$-algebras of $\Sigma$.

Let $\II$ and $\JJ$ be $\pi$-systems such that:

$\map \sigma \II = \GG$

and:

$\map \sigma \JJ = \HH$

where $\map \sigma \II$ and $\map \sigma \JJ$ are the $\sigma$-algebras generated by $\II$ and $\JJ$ respectively.

Then $\GG$ and $\HH$ are independent if and only if:

$\map \Pr {I \cap J} = \map \Pr I \map \Pr J$

for each $I \in \II$ and $j \in \JJ$.