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



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 $A$.
