Is there a notion of conditional probability with respect to a $0$ probability event when the event is more general than some variable equalling some value? The definition of conditional probability that I'm aware of relies on taking a limit, e.g. if $X$ is a continuous random variable, the conditional probability of $A$ given that $X=0$ is
$$\lim_{\epsilon \to 0^+}\frac{P(A \wedge -\epsilon < X < \epsilon)}{P(-\epsilon < X < \epsilon))}$$
But this doesn't have an obvious generalisation to conditional probabilities for other events.
A scenario I thought of is $X \sim U[0,1]$ and taking the probability of some event $A$ conditional on $X$ being in the Cantor set, $C$. E.g. intuitively, it seems you should be able to say that the probability that $X>\frac{1}{2}$ given that $X \in C$ is $\frac{1}{2}$. But the above definition doesn't allow a statement like this to be made, so I want to know if there is any more general notion of conditional probability which would allow this to make sense.
Going further, it intuitively seems like a statement like $\mathbb{E}[X | X \in C] = \frac{1}{2}$ should make sense.
Without a notion of conditional probability, it's unclear how to define independence, and it seems like it should be possible to define the independence of $0$ probability events. In particular, I've heard of infinite random graphs where whether each vertex exists has probability $p$ and is independent of every other vertex, and a given realisation of all other vertices has probability $0$, so this seems to be talking about conditional probability with respect to a $0$ measure set. So can conditional probability be defined in a way that statements like this make sense?
I've found this answer which seems to be talking about something similar, but I don't really understand how the conditional probability measure can be defined as a random variable, or how the definition they give can say anything about conditioning on $0$ probability events.