Let $(\Omega, \Sigma, P)$ be a probability space and $A, B \in \Sigma$ events. If $\Pr (B) = 0$ then there is no coherent definition for $\Pr (A | B)$. As Kolmogorov states, “the concept of a conditional probability with regard to an isolated given hypothesis [namely $B$ here] whose probability equals zero is inadmissible” [1, p.67]. This is the Borel-Kolmogorov paradox.
Kolmogorov resolved this paradox (at least to his own satisfaction -- the debate is still ongoing [2]) by giving a precise definition of conditional probability via the Radon-Nikodym theorem. The downside is that the Radon-Nikodym theorem does not define the probability conditioned on an event -- such as $\Pr (A | B)$ -- but rather it defines a probability $ \Pr (A | \mathcal F) \in L^1 (\Omega, \mathcal F, P)$ conditioned on a sub-$\sigma$-algebra $\mathcal F \subset \Sigma$.
It is my impression that, under a fixed sub-$\sigma$-algebra $\mathcal F$, there is a coherent definition for $\Pr (A | B)$ when $B \in \mathcal F$. That is, the Borel-Kolmogorov paradox does not arise if we fix the conditioning sub-$\sigma$-algebra $\mathcal F$. Is this true? If it is not true, then how could Kolmogorov claim to have resolved the paradox?
Secondly -- if it is true -- then how exactly is the paradox resolved? That is, how do we coherently define $\Pr (A | B)$ using $\Pr (A | \mathcal F)$?
I know that $$ \int_B \Pr (A | \mathcal F)(\omega) d P(\omega) = \Pr (A \cap B),$$ for any $B \in \mathcal F$. Further, if $B$ is an atom of $\mathcal F$, then $\Pr (A | \mathcal F)$ is a.e. constant on $B$ and equal to $\frac{\Pr (A \cap B)}{\Pr (B)}$. So I am able to recover the elementary definition of conditional probability $\Pr (A | B) = \frac{\Pr (A \cap B)}{\Pr (B)}$ from $\Pr (A | \mathcal F)$ when $\Pr (B) > 0$. But how can I derive a conditional probability $\Pr (A | B)$ from $\Pr (A | \mathcal F)$ when $\Pr (B) = 0$?
[1] Rao, M.M. Conditional Measures and Applications. Revised second edition. Boca Raton, FL: Chapman and Hall/CRC, 2005. https://doi.org/10.1201/9781420027433.
[2] Gyenis, Z., G. Hofer-Szabó, and M. Rédei. ‘Conditioning Using Conditional Expectations: The Borel–Kolmogorov Paradox’. Synthese 194, no. 7 (1 July 2017): 2595–2630. https://doi.org/10.1007/s11229-016-1070-8.
$\Pr (A | \mathcal F) $ does not coherently define $\Pr (A | B)$ when $P(B) = 0$. This is because the conditional expectation $\Pr (A | \mathcal F)$ is only defined almost everywhere. For example, suppose $\mathcal F$ is generated by a countable partition $\{B_i\}_{i=1}^\infty$ of $\Omega$. It is straightforward to prove that $$ \Pr (A | \mathcal F)(\omega) = \begin{cases} \frac{P(A \cap B_i)}{P(B_i)} & \text{ if } \omega \in B_i \text{ with } P(B_i) \ne 0, \\ x & \text{ if } \omega \in B_i \text{ with } P(B_i) = 0, \end{cases}$$ -- where $x$ is arbitrary -- is a version of $\Pr (A | \mathcal F)$. This version $\Pr (A | \mathcal F)$ gives a natural definition of the conditional probability $$\Pr (A | B_i) := \Pr (A | \mathcal F)(\omega),$$ where $\omega \in B_i$. This is consistent with the elementary definition of conditional probability for positive-probability events.
Now we can change $x$ and obtain another version of $\Pr (A | \mathcal F)$. But the conditional probability $\Pr(A | B_i)$ for zero-probability $B_i$ is different under this new version. So the conditional probability $\Pr (A | \mathcal F)$ on a sub-$\sigma$-algebra $\mathcal F$ cannot consistently define a conditional probability $\Pr (A|B)$ on a zero-probability event $B$. If we fix a version of $\Pr (A | \mathcal F)$, we can define $\Pr (A|B)$ -- but that definition is necessarily relative to our choice of version.
Thus, fixing a sub-$\sigma$-algebra $\mathcal F$ does not resolve Borel's paradox. We still get incoherent definitions of $\Pr (A | B)$ even when working under the same sub-$\sigma$-algebra.