A curious problem about Lebesgue measure. The Problem:
Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the open disc $B(0,n)$ (that is, the disc centered in (0,0) and with radius $n$).
Prove that if $\lim \inf \frac{\psi(n)}{n^{2}} = k > 0$, then there exists a ray starting from (0,0) that crosses an infinite number of the discs $(B(x_{m},0.5))_{m}$.
My Thoughts:
I find this problem particularly curious. There are several hints below the problem:


*

*Use that if $A \subset \mathbb{R}^{2}$ is Lebesgue-measurable and $k \geq 0$ then $kA= \{ kx:x \in A \} $ is Lebesgue-measurable too and $\lambda(kA)=k^{2}\lambda(A)$.

*Use that $\mu( \cup _{n} A_{n}) < +\infty$ implies $\mu( \lim \sup A_{n}) \geq \lim \inf _{n} \mu (A_{n})$ for any measure $\mu$.


I have thought about calling $R_{\alpha}$ to the ray with angle $\alpha$ and $A_{n} = \{ \alpha : R_{\alpha}$ crosses $B(x_{n},0.5) \}$. Then it would be enough to prove that $\lim \sup A_{n} \neq \emptyset$. Using the second hint, it is enought to prove that $\lim \inf \mu(A_{n}) >0$ for certain measure $\mu$.
It would be done if I could find a measure such that $\mu(A_{n})=\frac{\psi(n)}{n^{2}}$. I feel it is almost done but I'm stuck for nearly a week. Thanks in advance!

The idea is to rescale the disks contained in $B(0,n)$ by $n^{-1}$, thus obtaining a set contained in $B(0,1)$ with measure bounded from below. Some care must be taken to produce a sequence of such sets without double-counting disks.  
By the way, it suffices to assume that $$\limsup \frac{\psi(n)}{n^{2}} = k > 0$$ instead of $\liminf$.
Proof. Choose the sequence $n_j$ inductively, so that $\psi(n_1)>\frac{k}{2}n_j^2$ and for $j\ge 2$, $\psi(n_j)-\psi(n_{j-1})>\frac{k}{2}n_j^2$. Let $U_j$ be the union of disks that are contained in $B(0,n_j)$ but are not contained in $B(0,n_{j-1})$. The set $A_j:=n_j^{-1}U_j$ has Lebesgue measure at least $\frac{k}{2}$, and is contained in $B(0,1)$. Use the hint to conclude that the Lebesgue measure of $\limsup A_j$ is at least $k/2$. Any ray crossing $\limsup A_j$ crosses infinitely many disks.
