We know that conditional probability $P(A | B)$ is undefined when $P(B) = 0$. But this doesn't seem to be true to me always.
Consider the probability of chosing a real number between $r$ such that $0 \leq r \lt 1$ where any real number in $[0, 1)$ is equally likely to be chosen.
Thus, the sample space $S = [0, 1)$. Let $A = \{0.1\}$ and $B = \{0.1, 0.2\}$. Therefore, $P(A) = P(B) = 0$.
Now, speaking from intuition, if we are given that $r$ is either $0.1$ or $0.2$, one might conclude that the probability that $r = 0.1$ is $0.5$. Therefore, it seems like $P(A | B) = 0.5$. But this contradicts the fact that since $P(B) = 0$, $P(A|B)$ is undefined.
Where am I making a mistake?