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.


28D15 General groups of measure-preserving transformations
37A99 Ergodic theory
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