Let $\mathbb{Z}^2_{\mathrm{prim}}$ denote the set of vectors $(a,b) \in \mathbb{Z}^2$ such that $a$ and $b$ are relatively prime. Let $L>0$ and let $\nu_L^{\mathrm{prim}}$ denote the following counting measure on $\mathbb{R}^2$:
$$ \nu_L^{\mathrm{prim}} := \frac{1}{L} \sum_{v\in \mathbb{Z}^2_{\mathrm{prim}}} \delta_{\frac{1}{L}\cdot v}.$$
Exercise 2.3 on page 4 of this article, Counting Problems from the Viewpoint of Ergodic Theory: From Primitive Integer Points to Simple Closed Curves, asks the following:
Exercise 2.3: Show that the $SL(2,\mathbb{Z})$ orbit of the vector $(1,0)\in \mathbb{R}^2$ is precisely $\mathbb{Z}^2_{\mathrm{prim}} \subseteq \mathbb{Z}^2$. Conclude that any weak star limit point of the sequence $(\nu_L^{\mathrm{prim}})_{L>0}$ is $SL(2,\mathbb{Z})$ invariant.
I am able to see why the $SL(2,\mathbb{Z})$ orbit of $(1,0)$ is $\mathbb{Z}^2_{\mathrm{prim}}$. Why can we conclude from this that any weak star limit of the sequence $(\nu_L^{\mathrm{prim}})_{L>0}$ must be $SL(2,\mathbb{Z})$ invariant?