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 S∞ P∞ that µ( i=1 Ai) = i=1 µ(Ai) for any disjoint sets Ai ∈ 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 ∈ 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 ∞ f : X → R and |f| dµ < ∞ where p ∈ (0, ∞) and L consists of all p bounded functions to the reals. The space L (X, µ) has the norm kfkp = (R |f| dµ)1/p for p ∈ (0, ∞) and for p = ∞ is the supremum of |f|. 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 preserving transformation of a probability space (X, B, µ). If A is a measurable set, then for almost every x ∈ A there exists n ∈ N such that T n(x) ∈ A.

Proof. Let

k c [ −k B = {x ∈ A | T (x) ∈ A for all k ∈ N} = A − ( T (A). k∈N 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 measure and f : X → X be continuous and measure-preserving. Then L(f) = suppµ.

As a reminder the set L(f) = {x|x ∈ ω(x)} and

suppµ = {x ∈ X | µ(U) > 0 whenever x ∈ U and U is open}.

Proof. Since X is separable there exists a countable basis {Ui}i ∈ N for the topology of X. A point x is recurrent if for each Ui where x ∈ Ui there exists n an n ∈ N where f (x) ∈ 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˜ = {x | x ∈ ω(x)k and has full measure in X. 2