Lecture 11 Measure Theory and Recurrence February 7, 2008 1 Preliminaries Let X be a space and B a σ-algebra (i.e. closed under complements and countable unions of sets). A measure µ on B is a function to the reals such S1 P1 that µ( i=1 Ai) = i=1 µ(Ai) for any disjoint sets Ai 2 B. A set is a null set if µ(A) = 0. A set A has full measure if µ(Ac) = 0. A measure space (X; B; µ) is σ-finite if X can be expressed as a countable union of sets of finite measure. The σ-algebra B is complete if any subset of a null set is in B. We will assume that (X; B; µ) is a measure space where X is σ-finite and B is complete. If µ(X) = 1 we call µ a probability measure. A map between measure spaces is measurable if the preimage of any measurable set is measurable. A measurable map is non-singular if the preimage of any null set is a null set. Two measure spaces are isomorphic if there exists sets X0 ⊂ X and Y 0 ⊂ Y each of full measure and a map T : X0 ! Y 0 such that T is a bijection where T and T −1 are measurable maps. An isomorphism of a set to itself is called an automorphism. For a topological space X the smallest σ-algebra containing all open sets is called a Borel σ-algebra. A measure µ is Borel measurable if the measure of a compact set is finite. A point x 2 X is called an atom if µ(x) > 0. A finite measure space is a Lebesgue space if it is isomorphic to [0; a] with Lebesgue measure plus countably many atoms. A Lebesgue space with- out atoms is called non-atomic. 1 Two measurable functions are equivalent if they coincide on a set of full measure. The set Lp(X; µ) is the equivalence classes of functions such that R p 1 f : X ! R and jfj dµ < 1 where p 2 (0; 1) and L consists of all p bounded functions to the reals. The space L (X; µ) has the norm kfkp = (R jfj dµ)1=p for p 2 (0; 1) and for p = 1 is the supremum of jfj. The space L2(X; µ) is a Hilbert space with inner product < f; g >= R fg dµ. 2 Recurrence Theorem 2.1 (Poincar´eRecurrence Theorem) Let T be a measure preserv- ing transformation of a probability space (X; B; µ). If A is a measurable set, then for almost every x 2 A there exists n 2 N such that T n(x) 2 A. Proof. Let k c [ −k B = fx 2 A j T (x) 2 A for all k 2 Ng = A − ( T (A): k2N Then B is measurable and the T −k(B) preimages are disjoint and have the same measure. Therefore, µ(B) = 0 since T is measure preserving and µ(X) = 1. 2 Theorem 2.2 Let X be a separable metric space, µ a Borel probability mea- sure and f : X ! X be continuous and measure-preserving. Then L(f) = suppµ. As a reminder the set L(f) = fxjx 2 !(x)g and suppµ = fx 2 X j µ(U) > 0 whenever x 2 U and U is openg: Proof. Since X is separable there exists a countable basis fUigi 2 N for the topology of X. A point x is recurrent if for each Ui where x 2 Ui there exists n an n 2 N where f (x) 2 Ui. By the Poincar´eRecurrence Theorem there ~ ~ exists a set Ui ⊂ Ui of full measure such that every point in Ui returns to ~ ~ T Ui. Then the set Xi = Ui [ (X − Ui) has full measure in X. Let X = Xi. Then X~ = fx j x 2 !(x)k and has full measure in X. 2 2.