The Pointwise Ergodic Theorem and its Applications Peter Oberly 11/9/2018 Introduction Algebra has homomorphisms and topology has continuous maps; in these notes we explore the structure preserving maps for measure theory known (somewhat unimaginatively) as measure preserving transformations. The first section contains some (but not all) of the necessary definitions for this talk and in the second we introduce some classical examples to illustrate these definitions. We then turn our attention to the dynamics of measure preserving maps which leads us to the pointwise ergodic theorem. In the final section we use the ergodic theorem to prove Borel's theorem on normal numbers. Definitions Definition. A σ-algebra A is a collection of subsets of a non-empty set X so that X 2 A and A is closed under complementation and countable unions; The pair (X; A) is called a measurable space, and elements of A are called measurable sets. A particularly important σ-algebra is the collection of Borel sets, defined to be the σ-algebra generated by the open subsets of a topological space X. Definition. A measure m : A! [0; 1] is a function which satisfies the following: 1. m(E) ≥ 0 for all E 2 A; 2. m(;) = 0; 1 S P 3. If fEngn=1 ⊂ A is a sequence of pairwise disjoint sets in A, then m( n En) = n m(En). A measure space is a triple (X; A; m) where (X; A) is a measurable space and m is a measure defined on A. The triple (X; A; m) is called a probability space if m(X) = 1. Definition. Let (X; A; m) and (Y; B; n) be measure spaces, and let T : X ! Y be a map from X into Y . T is said to be measurable if T −1(E) 2 A for each E 2 B; that is, if the pre-image of every measurable set is measurable. Definition. A measurable transformation T :(X; A; m) ! (Y; B; n) is said to be measure- preserving if m(T −1(E)) = n(E) for all E 2 B. If T is a bijection and T −1 is also measure preserving, then T is said to be invertible. If (X; A; m) is a probability space, and if T : X ! X is measure preserving, then the quadruple (X; A; m; T ) is sometimes referred to as a measurable dynamical system. Remarks: (1) We should really write T :(X; A; m) ! (Y; B; n) since the measure preserving property Notes on the Ergodic Theorem depends on both the σ-algebras and the measures, but will often write T : X ! Y instead. (2) If T :(X; A; ; m) ! (Y; B; n) and S :(Y; B; n) ! (Z; C; p) are measure preserving, then so is S ◦ T . (3) Measure preserving maps are the structure preserving transformations (morphisms) of measure spaces. (4) As such, a measure preserving map T : X ! X induces a morphism on the Banach 1 1 1 space of m-integrable functions L (m). In detail, let UT : L (m) !L (m) be defined by UT (f) = f ◦ T . It is evident that UT is linear, and if f ≥ 0 (and so is real valued), then (UT f)(x) = f(T (x)) ≥ 0 for x 2 X. So UT is positive. In fact, UT is an isometry. For if Pn s is a non-negative simple function s = k=1 akχAk , where ak are scalars and Ak are the measurable sets where s > 0, then Z n Z n n Z X X −1 X UT (s) dm = ak χAk ◦ T dm = akm(T (Ak)) = akm(Ak) = s dm: k=1 k=1 k=1 Therefore choosing a sequence of simple functions sn which converges monotonically to jfj, 1 where f 2 L (m), shows jjUT (f)jj1 = jjfjj1. Note also that this shows UT really does map into L1(m). (5) As we are interested in the dynamics of measure preserving maps, from now on we will restrict our attention to measurable functions T : X ! X. Additionally, unless other wise stated, we will assume that (X; A; m) is a probability space. Our last definition requires a bit of motivation. Let (X; A; m; T ) be a measurable dynamical system. If T −1(E) = E for E 2 A, then T −1(X n E) = X n E and we could study our system by examining the two simpler systems (E; A\E; mjE A\E;T jE) and (X n E; A\(X n E); mjA\(XnE);T jXnE) (with the corresponding measures normalized appro- priately). If 0 < m(E) < 1, then we have actually decomposed our original system into two smaller ones. However, if m(E) = 0 or m(X n E) = 0 (i.e. m(E) = 1), then one of our simpler systems is in fact trivial, and we are left with a system essentially the same as the one we started with. It follows that those measurable dynamical systems where T −1(E) = E implies m(E) = 0 or 1 are not usefully decomposable in this way. It makes sense therefore to study those systems where such decomposition is not possible, for understanding these will enable us to understand the ones which can be simplified. We call such systems ergodic. Definition. A measurable dynamical system (X; A; m; T ) is said to be ergodic if E 2 A and T −1(E) = E implies that m(E) = 0 or 1. We will often have a specific probability space (X; A; m) in mind and refer to the measure preserving transformation T as ergodic. There are many characterizations of ergodicity; one which will prove useful in this talk is the following. Theorem 1. (X; A; m; T ) is ergodic if and only if f 2 L1(m) and f ◦ T = f ae implies that f is constant ae. Proof. Assume that for all f 2 L1(m) that if f ◦ T = f ae then f is constant ae. Let E 2 A −1 1 be so that T (E) = E. Then χE ◦ T = χE. As χE 2 L (m), then χE is constant ae. Therefore χE is either 0 or 1 ae and so m(E) = 0 or 1. The converse is more technical, and can be found in McDonald and Wiess on page 616. 2 Notes on the Ergodic Theorem Examples n n (1) Let T : R ! R be a linear map and let m be the Lebesgue measure on the Borel sets n n of R . If T is singular, then range T is a proper subspace of R , and so T is not measure pre- serving. If instead T is non-singular, from linear algebra then m(T −1(E)) = m(E)=j det T j for all Borel sets E. Therefore T is a measure preserving linear map if and only if j det T j = 1. (2) Let S1 = fz 2 C : jzj = 1g and let B denote the Borel σ-algebra. Then with normalized circular Lebesgue measure m, the triple (S1; B; m) is a probability space. For a 2 S1, define 1 1 the rotation Ta : S ! S by Ta(z) = az. Then Ta is measure preserving and invertible for all a. It is very instructive to show the following. Theorem 2. The rotation T = Ta is ergodic if and only if a is not a root of unity Proof. Suppose that a is a root of unity. Then ap = 1 for some p 6= 0. Let f : S1 ! S1 be defined by f(z) = zp. Then (f ◦ T )(z) = f(az) = apzp = f(z) for all z 2 S1. Therefore f ◦ T = f but f is non-constant. So T is not ergodic by theorem 1. Conversely let A be a measurable subset of S1 so that T −1(A) = A. Notice that the functions 1 1 n 2 en : S ! S defined by en(z) = z ; n 2 Z form an orthonormal basis for L (m). Let the P n Fourier series for χA be χA ∼ n bnen. Since en(T z) = a en(z), it follows by a change of variable that Z Z −n bn = χAe−n dm = a e−n dm; T −1(A) P n −1 and so χT −1(A) ∼ n a bnen. As T (A) = A, then χA = χT −1(A) and thus have the same n Fourier coefficients. Therefore bn = a bn for all n. If a is not a root of unity, the only way this can hold is if bn = 0 for all n 6= 0. By the uniqueness of Fourier coefficients, χA is a constant almost everywhere and so m(A) = 0 or 1. Therefore Ta is ergodic when a is not a root of unity. (3) Let ([0; 1); B; m) be the probability space consisting of the half open unit interval with Borel sets B and m the Lebesgue measure. Define T : [0; 1) ! [0; 1) by ( 2x; if 0 ≤ x < 1=2; T (x) = 2x mod 1 = 2x − 1; if 1=2 ≤ x < 1: This map is referred to as the dyadic transformation. Notice that if x has binary expansion x = 0:x1x2x3:::(2) then T (x) = 0:x2x3:::(2). It is worth showing that T is measure preserving. From measure theory [Billingsly, p 4], it suffices to prove that T preserves measure on a semi-algebra which generates the Borel σ-algebra. The collection of half open intervals with k j rational dyadic endpoints is such a semi-algebra. So let E = [ 2n ; 2n ) where n ≥ 0 and 3 Notes on the Ergodic Theorem 0 ≤ k ≤ j ≤ 2n.