On a measurable analogue of small topological full groups

In the case of a measure-preserving ergodic Z-action, the closure of the derived group is shown to be the kernel of the index map. If such an action is moreover by homeomorphism on the Cantor space, we show that the topological full group is dense in the L1 full group. Using Juschenko-Monod and Matui's results on topological full groups, we conclude that L1 full groups of ergodic Z-actions are amenable as topological groups, and that they are topologically finitely generated if and only if the Z-action has finite entropy.