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ergthm - Birkho Ergodic Theorem Jonathan L.F King...

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Birkhoff Ergodic Theorem Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 23 November, 2009 (at 01:24 ) Pointwise convergence Fix an mpt ( T : X, X , μ ) on a prob.space. Let A N ( f ) denote the fnc whose value at x is A k N f ( T k x ) . Use E ( · | · ) for the conditional ex- pectation operator. The proof below is a symmetrized version of the Katznelson-Weiss proof. 1 : Birkhoff Ergodic Theorem. Fix f L 1 ( μ ) . Then this almost-everywhere limit exists, a . e-lim N →∞ A N ( f ) = E ( f | I ) , where I denotes the field of T -invariant sets. Reduction. ( The crux is proving a.e-convergence. For then, identifying the limit function as E ( f | I ) is not difficult. ) Let f and f denote the pointwise lim[sup , inf] N →∞ A N ( f ). It suffices to show that Z f 6 Z f . 2 : For applying this to f yields R f 6 R f . Hence R f 6 R f 6 R f . Since R f 6 R f yet f () > f () , it follows that f a.e = f . Let f M denote Max( f, M ), for M Z - . So f + > f 1 > f 2 > f 3 > . . . , and the Monotone Convergence Thm , implies R f M & R f as
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