Probability measure domain

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.

• I'm afraid I don't see the question about independence and other probability spaces: what does the domain of $P$ have to do with that? Clarify a bit more? Give an example? Feb 18 '19 at 22:28
• Apologies, I'll edit my question Feb 18 '19 at 22:40
• Ok I edited my question Feb 18 '19 at 22:58
• @JasonM the sigma fields $\mathscr{F}_i$ are contained in $\mathscr{F}$ (so $\mathscr{F}$ is at least as large as $\sigma(\cup \mathscr{F}_i)$). Feb 18 '19 at 23:00
• Without more context (I don't have the book) it also seems confusing to me. One should expect that $P$ is defined on the $\sigma$-field generated by all $\mathcal{F}_i$. Feb 18 '19 at 23:03

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.
• The relation is that the sigma fields $\mathscr{F}_i$ are subset of $\mathscr{F},$ (otherwise the independence is a nonsensical issue). Feb 18 '19 at 23:03
Thanks to Will M. it appears the most sensible $$\sigma-$$field for $$P$$ to be defined on is a $$\sigma-$$field containing $$\sigma(\cup_{i=1}^n \mathcal{F}_i)$$. Without the experience, I had difficulty seeing this, so thanks to everyone who commented.