Invariant Radon measures and minimal sets for subgroups of $text{Homeo}_+(mathbb{R})$

Let $G$ be a subgroup of $text{Homeo}_+(mathbb{R})$ without crossed elements. We show the equivalence among three items: (1) existence of $G$-invariant Radon measures on $mathbb R$; (2) existence of minimal closed subsets of $mathbb R$; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup $G$ of $text{Homeo}_+(mathbb{R})$, we show that $G$ always has an invariant Radon measure and a minimal closed set if every element of $G$ is $C^{1+alpha} (alpha>0$); a counterexample of $C^1$ commutative subgroup of $text{Homeo}_+(mathbb{R})$ is constructed.