This was an interesting problem that came up in qual studying.
$SL(2, \mathbb{R})$ acts on $\mathbb{R}^2$, and hence on any measure on $\mathbb{R}^2$. What are the Baire measures (i.e. Borel measures finite on compact sets) that are invariant under this action?
Two obvious ones are the Lebesgue measure, and delta masses at the origin. I suspect that there are none which are not a linear combination of these. Intuitively, it seems like if we can find some set which has very small Lebesgue measure, then we should be able to squeeze many $SL(2, \mathbb{R})$-orbits of it into a compact set, but I haven't been able to formalize this.