Contractibility of the Automorphism Group of a Nonatomic Measure Space

It is shown that the group of automorphisms of a nonatomic finite measure space is contractible. 1. In [l] S. Harada showed that the automorphism group of the unit interval with Lebesgue measure is simple and arcwise connected. Our purpose here is to give a simple proof that the automorphism group G of a nonatomic measure space (ß, (B, m) with rez(ß)< oo is contractible. The contraction map will be constructed directly by using induced transformations. Let B be the measure algebra of (Q, 03, m); B is a metric space with the metric |E, F\ = m(EAF) (E, FEB). An automorphism TEG maps B onto B isometrically. We shall consider both the strong topology on G (induced by uniform convergence on B) and the weak topology on G (induced by pointwise convergence on B). With either of these topologies, G is a topological group. 2. For EEB and TEG we set Oo = Ec, (1) «i = E C\ T~lE, ak = e r\ t-ie° r\ ■ ■ ■ r\ t-o-ve' r\ T-kE (k ̂ 2). By Poincaré's recurrence theorem, {ñ0, flii • • • } forms a partition of ñ called the return partition of E and T. Now we define the automorphism TE induced by E and T by TB(u) = Z*(w) (» G Ö*, & » 0,1, • • • ). It is not hard to show that TeEG. 3. Lemma. The map G defined by 0. Then there exists a positive integer kg such that Received by the editors February 3, 1970. AMS 1968 subject classifications. Primary 4750; Secondary 2870.