What does it mean for a Poisson point process $\Phi$'s points in $A$, conditioned on $\Phi(A)=k$ to be uniform?

Question

I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these points are uniformly distributed in $A$.

I'm having trouble making sense of what this means. "Conditional on $\Phi(A)=k$ I guess means consider the process $\Phi 1_{\Phi(A)=k}$ and then divide probabilities by $P(\Phi(A)=k)$. But, probabilities of what exactly? How am I labeling the points $X_1,\ldots,X_k$? In $\mathbb{R}$ If I did so by $X_1< X_2 < \cdots < X_k$ then clearly they are not uniformly distributed, so clearly the way that I label them matters. Hence my question, what is meant by saying $X_1,\ldots,X_k$ are uniformly distributed?

Answer 1

For every measurable set $A\subseteq\mathbb R^d$ of finite measure, and every measurable set $B\subseteq A$, let $p$ be the conditional probability that the number of sites in $B$ is $\ell$, given that the number of of sites in $A$ is $k$.

Suppose $X_1,\ldots,X_k\sim\text{i.i.d. Uniform}(A)$. Let $q$ be the probability that $|\{ X_1,\ldots,X_k \} \cap B| = \ell$.

Then, regardless of which sets are $A$ and $B$ and which numbers are $k$ and $\ell$, we have $p=q$.

In other words, the probability distribution of the number of points falling in $B$ given the number in $A$, is always the same in either of those two scenarios.

Answer 2

I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these points are uniformly distributed in $A$.

$\Phi$ is a process which generates iid distributed points in $\mathbb{R}^d$, then if $A \subseteq \mathbb{R}^d$ with $k$ points, then the $k$ points within $A$ are uniformly distributed in $A$.