Determining the distance of Poisson-distributed stars in $\mathbb{R}^3$. I want to solve the following exercise:

Suppose the distribution of the star in space is a
  Poisson-distribution, i.e. the probability that there are $n$ stars in
  a region $T\in\mathbb{R}^{3}$ is $e^{-\lambda}\frac{\lambda^{n}}{n!}$,
  where $\lambda$ is proportional to the volume of $T$. We choose
  randomly a point $x\in\mathbb{R}^{3}$. Let $X$ be the distance of $x$
  to the next star. Then calculate the probability distribution function
  of $X$, i.e. calculate $P\left(s\leq X\leq t\right)$.

My questions are:
1) Is this problem even well-defined ?
Saying that stars in some region
$T\in\mathbb{R}^{3}$ are Poisson-distributed (shortly: "P-distr.")
confuses me, since $T$ is not fixed, so actually I have for every
possible volume a different P-distr.. Thus, if the next star is for
example within distance $A$ of $x$ I can have many differently shaped
$T$'s with different volumes that contain $x$ at the " center"
(whatever that is) and that star, so I have different P-distributions
that measure my distance (Intuitively suppose I should take balls
around $x$, but this explanation is not rigorous).
Conclusion: Not having a "fixed unit" $T$ with which to
measure distance, makes this problem not well-defined?
2) What is the image of $X$ ? Heck, what is even our probability
space ?
Possible (but very unsure) explanation: Since we dealt only with discrete
(countable) probability models so far, I assume that we somehow have
to approximate $\mathbb{R}^{3}$ by $\mathbb{Q}^{3}$ or $\mathbb{Z}^{3}$
(which are still countable...) and take that as our probability model
$\Omega$ (I think that by symmetry we could assume $x$ to be the
origin) and take our $X:\mathbb{Q}^{3}\rightarrow\mathbb{R}$ as mapping
$\left(x,y,z\right)\mapsto\sqrt{x^{2}+y^{2}+z^{2}}$. This explanation
would at least coincide with the fact that this exercise asks only
for $P\left(s\leq X\leq t\right)$ instead of precisely $P\left(X=q\right)$
(although the wording " distribution" would rather mean the
latter, I think...), since if have approximated the exact location
of the star by a rational number, to make the theory work. But this
seems also sketchy to me, since I don't know how to approximate the
error (what should $s,t$ be ?)
(Or, a different line of thought; we accept only rational numbers
as distances; but what would $\Omega$ the be ?)
 A: 
1) Yes. $\ \ $ 2) $\mathbb R_+$.

A possible choice of probability space $(\Omega,\mathcal F)$ is as follows:


*

*each $\omega$ in $\Omega$ is a locally finite subset of $\mathbb R^3$,

*$\mathcal F$ is the smallest sigma-algebra such that counting functions of Borel subsets are measurable.


In other words, one asks that $[N_B=k]$ is in $\mathcal F$, for every Borel subset $B$ of $\mathbb R^3$ and every integer $k$, where $N_B(\omega)=\#(\omega\cap B)$.
Then $X:(\Omega,\mathcal F)\to(\mathbb R_+,\mathcal B(\mathbb R_+))$ is a random variable since $[X\gt r]=[N_{B(r)}=0]$ where, for every real number $r\gt0$, $B(r)$ is the ball in $\mathbb R^3$ with radius $r$ and centre $x$.
And $\mathrm P(N_{B(r)}=0)$ is not difficult to compute since one knows the distribution of $N_{B(r)}$.
A: 1) Yes, it's well defined. You should see it if your are familiar with analogous Poisson  process in the real line. In that case, you have a point process with "intensity" $\alpha$, so that if you take a very small interval $dx$ the probability that an event occurs in that interval is $p = \alpha \; dx$, and the events ocurrences in different intervals are independent. Hence, letting $dx \to 0$ you get that the number of events in any interval of length $L$ (actually, in any measurable subset of $\mathbb{R}$) is a Poisson variable with expected value $\lambda = \alpha L$. The same happens here, in $\mathbb{R}^3$.
2) You must pick a point at random in $\mathbb{R}^3$ and measure the distance to the nearest event ("star"). $X$ takes values in $(0,+\infty)$, and it's not difficult to compute, considering that $P(X \ge x)$ is just the probability that a ball of radius $x$ has zero events.
