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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:
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