Suppose we are picking points uniformly at random from the surface of the Earth. I want to compute the probability that I pick a point in the Western hemisphere, given that I pick a point on the equator. The answer should clearly be $1/2$.
From the definition, we have
where event $A$ is choosing a point in the Western hemisphere and $B$ is choosing a point on the equator. As the equator is a $1$-dimensional smooth line on a $2$-dimensional surface, it has measure $0$. So I compute $P(B)=0$. But using the conditional probability formula requires $P(B)>0$. In fact, this is the definition of conditional probability! So how do I make sense of $P(A|B)$? Clearly, it should work out to be $1/2$, but what is the rigorous way to compute it?