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

中文翻译：

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Homeo+(R) 子群的不变氡测度和最小集

摘要 设 G 是 Homeo + ( R ) 的一个没有交叉元素的子群。我们展示了三个项目之间的等价性：（1）在 R 上存在 G 不变的氡测量；(2) R 的最小闭子集的存在性；(3) 不存在覆盖全线的无限塔。对于 Homeo + ( R ) 的幂零子群 G，如果 G 的每个元素都是 C 1 + α ( α > 0 )，我们证明 G 总是有一个不变的 Radon 测度和一个最小闭集；构造了Homeo + ( R ) 的C 1 交换子群的反例。