$\{A_i, i \in \mathbb{N} \}$ are defined to be independent, if $P(\cap_{k=1}^{n} A_{i_k}) = \prod_{k=1}^{n} P(A_{i_k}) $ for any finite subset of $\{A_i, i \in \mathbb{N} \}$.
- We know $P(\cup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i) $ iff $\{A_i , i \in \mathbb{N}\}$ are disjoint, which is independent of the probability measure and purely depends on the relation between the sets. I was wondering if it is possible to similarly characterize/interpret $\{A_i , i \in \mathbb{N}\}$ being independent purely from relation between sets, and make it independent of the probability measure as much as possible if completely is impossible?
- Is the definition of $\{A_i, i \in \mathbb{N} \}$ being independent equivalent to $P(\cap_{i=1}^{\infty} A_{i}) = \prod_{i=1}^{\infty} P(A_{i}) $. What is the purpose of considering any finite subset instead?
Is generalization of independence from probability space to general measure space meaningful?
The only interpretations of independence I know are: measure can be exchanged with product/intersection on independent sets, and intuitively, independent events occur independently of each other. Are there other interpretation, especially in the general measure space setting?
Thanks and regards!