In more mathematical settings, you can successfully condition on events with probability 0 (for instance, if (X,Y) follow a bivariate normal distribution, you might want to know the probability distribution of Y given X=x).

You can't really do this, since the answer depends on how you take the limit. You can find a limit of conditional probabilities, but saying "the probability distribution of Y given X=x" is ambiguous. This is known as the Borel-Kolmogorov paradox.