Existence of random points

Question

Let $\mathbb{S}^k$ be the $k$-dimensional unit sphere and let $\sigma(\mathbb{S}^k)$ be its surface area. Suppose we have a regular area partition $\{S_i\}_{i=1}^n$ of $\mathbb{S}^k$ with constant $c>0$, i.e., $$\mathbb{S}^k = \bigcup_{i=1}^n S_i,$$ with $\{S_i\}_{i=1}^n$ pairwise disjoint and $$\sigma(S_i) = \frac{\sigma(\mathbb{S}^k)}{n},\quad \text{diam}(S_i)\leq cn^{-\frac{1}{k}},$$ for all $i=1,\ldots,n$.

Let $\{\xi_i\}_{i=1}^n$ be random points (or random variables), such that $\xi_i \in S_i$ and $$\mathbb{P}(\xi_i \in A) = \frac{\sigma(S_i\cap A)}{\sigma(S_i)},$$ for all $i=1,\ldots,n$ and for all measurables sets $A\subset\mathbb{S}^k$.

In the paper I am studying the author claims, and gives no further detail or proof, that such a collection of points exists and, moreover, that we can assume that they are independent. How can I justify these two things?

Answer 1

For each $i$, the triple $(S_i, \mathcal{F}_i, P_i)$ is a probability space where:

• $\mathcal{F}_i= \{S_i \cap A \mid A\subseteq\mathbb{S}^k\text{ is measurable}\}$
• $P_i(S_i \cap A) := \frac{\sigma(S_i \cap A)}{\sigma(S_i)}$.

Then the identity function $\xi_i:S_i \to S_i$ is a random point having the desired distribution.

Note that a collection of random variables having the same marginal distributions as the $\xi_i$'s are not necessarily independent. However, there exist independent copies of the $\xi_i$'s having the desired marginal distributions. These random variables can be defined on the probability space $(\prod_{i=1}^n S_i, \mathcal{F}, P)$, where

• $\mathcal{F} = \{(S_1 \cap A_1,\dots,S_n \cap A_n) \mid A_1,\dots,a_n \subseteq \mathbb{S}^k \text{ are measurable}\}$
• $P(S_1 \cap A_1,\dots, S_n \cap A_n) := P_1(S_1 \cap A_1)\cdots P_n(S_n \cap A_n)$.

Then just take $\xi_i:\prod_{i=1}^n S_i \to \mathbb{S}^k$ to be the projection $\xi_i(s_1,\dots,s_n)=s_i$.