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Another rather nice example: the group of automorphisms of $[0,1]$ endowed with the Lebesgue measure (i.e, bi-measurable maps preserving the measure, identified when they coincide outside a set of Lebesgue measure zero). In a related vein, the full group of any countable, ergodic, probability measure preserving Borel equivalence relation on $[0,1]$ is simple.