I've been studying a course in probability theory, and I'm getting tied up with the formal treatment of conditional probability. Breiman explains quite well that the usual definition,
$$ \mathbb{P}(A \;|\; B) := \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} $$
only makes sense if $\mathbb{P}(B)>0$, and sometimes we want to condition on sets of measure zero, eg. $\mathbb{P}(A \;|\; X=x)$ over some continuous state space. So far, so good. In order to deal with this problem, we instead use an (in my opinion) long-winded definition that comes from Radon-Nikodym derivatives, and defines the conditional probability in terms of the conditional expectation, and leads to the problem of regular conditional probabilities which may or may not exist and are only almost surely unique. In my opinion it's a total mess, but it's worth it if it means you can get a rigorous treatment.
But then I discovered the Borel-Kolmogorov paradox, which shows us that conditioning on a set of measure zero isn't even well-defined, and to the best of my knowledge, Breiman's treatment (which I understand was devised by Kolmogorov and is considered the standard) doesn't help here. So my question is twofold:
- Does the formal approach actually solve the paradox? I expect the answer here is no, but I'd be happy to be proven wrong.
- If not, what is the point of the formal treatment if it doesn't actually allow us to do the one thing it's set up to do -- namely, to condition on events with probability zero?