This question arises as a possible step in answering this unsolved question on MSE.
Assume a random vector $v$ is sampled uniformly from the unit sphere in $d$ dimensions, $v \in S^{d-1} \subset \mathbb{R}^d$. Assume we are given a finite collection of 'open rectangular' subsets $V_i \subset S^{d-1}$ in the sense of being written as $$ V_i = (x_i^1, y_i^1)\times \cdots \times (x_i^{d-1}, y_i^{d-1}) $$ for spherical coordinates $x_i^1 < y_i^1 \in (-\pi, \pi]$ and $x_i^j < y_i^j \in (-\pi/2, \pi/2]$ for $j\geq 2$. Assume we are given these $(x_i^j, y_i^j)$ explicitly. Then it is not so difficult to find an explicit formula for $P(v \in V_i)$ for any fixed $i$. I am looking for a way to compute $$P(v \in \cup_i V_i)$$ when the sets $V_i$ are not disjoint (otherwise one could simply sum the probabilities of individual membership). This could either be an exact formula or an algorithmic procedure. I am running into the issue that the union of rectangular regions is no longer rectangular, so the probability is no longer easy to compute. Consider for example $d=3$ and \begin{align} V_1 = (-\pi, \pi) \times (0, \pi/2) \\ V_2 = (0, \pi) \times (-\pi/2, \pi/2) \\ \end{align} i.e. $V_1$ is the upper hemisphere and $V_2$ the 'right' hemisphere. Then $$P(v \in V_1) = P(v \in V_2) = 1/2$$ and in this case we can figure out that the union can be rewritten as a disjoint union $$V_1 \cup V_2 = \left((-\pi, 0) \times (0, \pi/2)\right) \cup \left((0, \pi) \times (-\pi/2, \pi/2)\right)$$ to obtain $$P(v\in V_1 \cup V_2) = P(v \in (-\pi, 0) \times (0, \pi/2)) + P(v \in (0, \pi) \times (-\pi/2, \pi/2)) = 1/4 + 1/2 = 3/4 \,,$$ but I'm unsure how to do this in general. Any help appreciated!
