A descriptive set theorist's proof of the pointwise ergodic theorem
aa r X i v : . [ m a t h . D S ] J un A DESCRIPTIVE SET THEORIST’S PROOF OF THE POINTWISE ERGODIC THEOREM
ANUSH TSERUNYAN
Abstract.
We give a short combinatorial proof of the classical pointwise ergodic theorem for probabilitymeasure preserving Z -actions [Bir31]. Our approach reduces the theorem to a tiling problem: tightlytile each orbit by intervals with desired averages. This tiling problem is easy to solve for Z with intervalsas tiles. However, it would be interesting to find other classes of groups and sequences of tiles for whichthis can be done, since then our approach would yield a pointwise ergodic theorem for such classes. Let (
X, µ ) be a standard probability space and f ∈ L ( X, µ ). For a finite nonempty U ⊆ X , put A f [ U ] .. = 1 | U | X y ∈ U f ( y ) , and for a finite equivalence relation F on X , define A f [ F ] : X → R by A f [ F ]( x ) .. = A f h [ x ] F i . Lemma 1 (Finite averages) . For any measure-preserving finite equivalence relation F on ( X, µ ) , Z f dµ = Z A f [ F ] dµ. Proof.
For each n ∈ N , restricting to the part of X where each F -class has size n , we may assume X isthat part to begin with. Because each F -class is finite, there is a Borel automorphism T that induces F and a Borel F -transversal B ⊆ X . Using the invariance of µ , we deduce Z X f ( x ) dµ ( x ) = Z B X i
Classical Descriptive Set Theory , Grad. Texts in Math., vol. 156, Springer, 1995.[KM04] A. S. Kechris and B. Miller,
Topics in Orbit Equivalence , Lecture Notes in Math., vol. 1852, Springer, 2004.(Anush Tserunyan)
Department of Mathematics, University of Illinois at Urbana-Champaign, IL, 61801, USA
E-mail address : [email protected] Although, instead, we could easily observe that these sets are in the σ -ideal generated by analytic sets and use thatthese sets are universally measurable.-ideal generated by analytic sets and use thatthese sets are universally measurable.