Borel's Paradox: Conditioning on Measure-Zero Events

Let $(\Theta, \Phi)$ be uniformly distributed on the unit sphere $S^2$, where $\Theta \in [0, 2\pi)$ is the longitude and $\Phi \in [0, \pi]$ is the colatitude, with joint density $f(\theta, \phi) = \frac{1}{4\pi} \sin \phi$.

(a) Compute the conditional distribution of $\Phi$ given $\Theta = 0$ by treating $\Theta$ as the conditioning variable (i.e., compute $f(\phi \mid \theta = 0)$).

(b) Now reparameterize: let $X = \cos(\Theta) \sin(\Phi)$, $Y = \sin(\Theta) \sin(\Phi)$, $Z = \cos(\Phi)$. The great circle $\{\Theta = 0\}$ can equivalently be described as $\{Y = 0, X \geq 0\}$. Compute the conditional distribution of $\Phi$ given $Y = 0$ and $X > 0$. Is it the same as in part (a)?

(c) Explain why the two answers differ. What does this tell us about the meaning of 'conditioning on a measure-zero event,' and what is the correct mathematical framework for resolving this ambiguity?