On a measurable space $(\Omega, F)$, where $\Omega$ is a set of outcomes and $F$ is a $\sigma -$field, what exactly is the domain of a probability measure $P$? If it's a specific $\sigma -$field such as $F$, then how can we define the independence on sets from multiple spaces? And it can't be the whole power set because we can construct sets that have no probability. I thought the whole point of using a $\sigma -$field was to make the domain of probability measures better to define.
Edit:
So in the textbook $Probability: \ Theory \ and \ Examples$, by Rick Durrett, he says "... $\sigma -$fields $\mathcal{F}_1, \ldots , \mathcal{F}_n$ are independent if whenever $A_i \in \mathcal{F}_i$ for $i=1, \ldots, n$, we have $$P(\cap_{i=1}^{n} A_i)= \prod_{i=1}^n P(A_i)$$..."
How can $P$ be defined on all these $\sigma-$fields? I thought a probability measure was always w/respect to a given $\sigma-$field.
The domain of $P$ is definitely $F$, the $\sigma$-field on $\Omega$ (it is in some cases the power set of $\Omega$, e.g. for discrete measures on at most countable $\Omega$).
$P$ a function from $F$ to $[0,1]$ obeying certain axioms. I don't see a relation to independence.