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New Borel050707 - 1 NEW BOREL INDEPENDENCE RESULTS by...

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1 NEW BOREL INDEPENDENCE RESULTS by Harvey M. Friedman April 30, 2007 May 7, 2007 sketch 1. Introduction. 2. Countable Borel equivalence relations on . 3. Countable Borel equivalence relations on 2 . 4. Degrees into finitely generated groups. 5. Borel functions on groups. 6. Countable Borel quasi orders. 1. INTRODUCTION. A considerable amount of interdisciplinary work has been done on countable Borel equivalence relations. See [JKL02]. Many researchers in logic have worked on countable Borel equivalence relations, who have utilized the work of many people in ergodic theory and related areas. These include the following names taken from the list of references of [JKL02]: S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, S. Thomas, A. Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M. Wright, R.J. Zimmer. Many thanks to Simon Thomas for advice on this manuscript. A Borel equivalence relation is a pair (X,E), where X is a Polish space (complete separable metric space), and E is a Borel equivalence relation on X. A countable Borel equivalence relation is a Borel equivalence relation in which every equivalence class is countable.
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2 A fundamental situation is that of a Borel function f:X which is E invariant. This means that for all x,y X, E(x,y) f(x) = f(y). In this note, we will be concerned with the following kind of result. Let (X,E) be a countable Borel equivalence relation. Every E invariant Borel function f:X is constant on a "big set". In the case where X is also equipped with a measure mu, this statement with "big set" = set of full measure is known as (X,E, m ) is ergodic. Note that because of the countable additivity, ergodicity has these three equivalent forms. Every E invariant Borel function f:X is constant on a set of full measure. Every E invariant Borel function f:X {0,1} is constant on a set of full measure. Every E invariant Borel subset of X contains or is disjoint from a set of full measure. We can fix a very standard probability space like I n with Lebesgue probability measure or R n with Lebesgue measure or {0,1} with the usual probability measure. Then we can search for necessary or sufficient conditions for a countable Borel equivalence relation E on I n , R n , {0,1} , respectively, to be ergodic. We mention three well known facts about ergodicity. it won't make any difference which standard Borel space we choose. We will simply choose R for convenience.
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3 ERGODIC 1. There is a countable Borel equivalence relation (R,E) which is ergodic. In particular, E(x,y) iff x-y is rational.
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