What is the broadest definition of conditional probability? The usual definition of conditional probability on a probability space $(\Omega, \sigma, P)$ is~:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
which obviously implies that $B$ must not be a negligible set. However this definition appears too restrictive as we sometimes need to compute probabilities conditioned by negligible sets (a usual case is for example an event of the form $B = \{X = x\}$ where $X$ is a continuous random variable and $x$ a real number). Informally speaking we would like to define~:
$$P(A|B) = \underset{P(B') > 0}{\lim_{B'\rightarrow B}} \frac{P(A \cap B')}{P(B')}$$
Now the meaning of the limit should be made precise. I'm neither aware of a topology defined on $\sigma$ itself nor convinced it is the way to go. It might also be possible to work with sequences of decreasing events which in some sense "convergence" toward $B$:
$$B_1 \supset B_2 \supset \cdots \supset B$$
such that:
$$\forall x \in \Omega \setminus B, \exists n \in \mathbb N,  x \not\in B_n$$
But I'd like to know if some formal definition already exist.
 A: The conditional probabilities used in modern mathematical probability theory are defined with respect to a $\sigma$-algebra.
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{G}$. Then the conditional probability $P(\cdot \mid \mathcal{G})$ is a $\mathcal{G}$-measurable random variable that satisfies
$$P(\cdot \cap G) = \int_G P(\cdot \mid \mathcal{G})dP$$
for all $G \in \mathcal{G}$. The existence and almost sure uniqueness of $P(\cdot \mid \mathcal{G})$ is guaranteed by the Radon-Nikodym theorem. If you're familiar with conditional expectations, you'll notice that, for $A \in \mathcal{F}$, we have $P(A \mid \mathcal{G}) = E(\mathbf{1}_A \mid \mathcal{G})$.
This generalizes the definition of conditional probability that you mention as follows. Let $\mathcal{G}$ be generated by a partition $\{E,E^c \}$ with $P(E)>0$. Then, since $P(\cdot \mid \mathcal{G})$ is $\mathcal{G}$-measurable, it is constant on $E$ (this being an atom of $\mathcal{G}$). Therefore, by the equation above we have, for almost all $\omega \in E$,
$$P(\cdot \mid \mathcal{G}) = \frac{P(\cdot \cap E)}{P(E)} = P(\cdot \mid E).$$
We can use this theory of conditional probability to condition on probability $0$ events by first embedding such events in a $\sigma$-algebra. For examples, see any of the standard textbooks on measure-theoretic probability, e.g. Billingsley.
There are other theories of conditional probability in which conditional probabilities are primitive, and not defined in terms of unconditional probabilities as we've done above. For example, Alfred Renyi developed such a theory and it is explained reasonably clearly in chapter 4 of Rao's Conditional Measures and Applications. As far as I can tell, Renyi's theory hasn't been nearly as influential as the theory explained above.
