There is a very general fact from a paper I am reading which I don't know the proof:
Let $X=G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$. Let $\mu$ be a $G$-invariant probability measure on $X$. For a subsemigroup $F$ of $G$ which acts on $X$ ergodically, let
$$S:=\{x \in X: F.x ~\text{not dense in}~ X\}$$
then $S$ is a $\mu$-null set.
If $F$ is a group then I know a proof by mimicking the argument here by putting $S$ in a countable union of null sets.
I wonder how to prove this fact in general or it follows from some ergodic theorems?
Source (first page of)