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On the cost of generating an equivalence relation. (English) Zbl 0843.28010
In this paper a generating system for a measurable equivalence relation \(R\) is a countable family \(\Phi\) of partially defined \(\mu\)-preserving isomorphisms \(\varphi: A_j\to B_j\) \((A_j, B_j\subset X)\) on a standard Borel space \(X\), so that \(R\) is the smallest equivalence relation generated by the \(\varphi_j\)’s. It is shown that \[ \sum_j \mu(A_j)+ \int {1\over \# R(x)} \mu(dx)\geq 1\quad (\text{where}\quad {1\over \infty}= 0) \] and equality holds if and only if \(R\) is amenable and the generators are independent. Applications to pseudogroups of measure-preserving homeomorphisms are also given.

MSC:
28D15 General groups of measure-preserving transformations
37A99 Ergodic theory
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References:
[1] Gaboriau, Dynamique des systèmes d’isométries et actions de groupes sur les arbres réels (1993)
[2] Feldman, Trans. Amer. Math. Soc. 234 pp 289– (1977)
[3] DOI: 10.1017/S014338570000136X · Zbl 0491.28018
[4] DOI: 10.1017/S0143385700005368 · Zbl 0667.28003
[5] Salem, Riemannian Foliations, Progress in Math. 73 (1988)
[6] DOI: 10.2307/2118526 · Zbl 0843.57026
[7] Haefliger, Structures Transverses des Feuilletages 116 pp 70– (1984)
[8] DOI: 10.1007/BF01391835 · Zbl 0594.57014
[9] DOI: 10.1017/S0143385700008580 · Zbl 0839.58022
[10] DOI: 10.1007/BF02773004 · Zbl 0824.57001
[11] DOI: 10.1007/BF01244321 · Zbl 0791.58055
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