Institute of Mathematics • Freie Universitat¨ Berlin • Peter´ Koltai ERGODIC THEORY AND TRANSFER OPERATORS —HANDOUT 1 — Summer 2015 Measure theory and Lebesgue integration 1 Measures and measure spaces 1.1 Basic definitions and properties We collect the most basic definitions in measure theory, followed by some results which will be useful in the lectures. Definition 1 (Algebra and s-algebra ): Consider a collection A of subsets of a set X, and the following properties: (a) When A 2 A then Ac := X n A 2 A. (b) When A, B 2 A then A [ B 2 A. S (b’) Given a finite or infinite sequence fAkg of subsets of X, Ak 2 A, then also k Ak 2 A. If A satisfies (a) and (b), it is called an algebra of subsets of X; if it satisfies (a) and (b’), it is called a s-algebra . It follows from the definition that a s-algebra is an algebra, and for an algebra A holds • Æ, X 2 A; • A, B 2 A ) A \ B 2 A; • A, B 2 A ) A n B 2 A; T • if A is a s-algebra , then fAkg ⊂ A ) k Ak 2 A. Definition 2 (Measure): A function m : A! [0, ¥] on a s-algebra A is a measure if (a) m(Æ) = 0; (b) m(A) ≥ 0 for all A 2 A; and S (c) m ( k Ak) = ∑k m(Ak) if fAkg is a finite or infinite sequence of pairwise disjoint sets from A, that is, Ai \ Aj = Æ for i 6= j. This property of m is called s-additivity (or countable additivity). If, in addition, m(X) = 1, then m is called a probability measure. Definition 3: (a) If A is a s-algebra of subsets of X and m is a measure on A, then the triple (X, A, m) is called a measure space. The subsets of X contained in A are called measurable. (b) If m(X) < ¥ (resp. m(X) = 1) then the measure space is called finite (resp. probabilistic or normalized). S (c) If there is sequence fAkg ⊂ A satisfying X = k Ak and m(Ak) < ¥ for all k, then the measure space (X, A, m) is called s-finite. A set N 2 A with m(N) = 0 is called a null set. If a certain property involving the points of a measure space holds true except for a null set, we say the property holds almost everywhere (we write a.e., which, depending on the context, sometimes means “almost every”). We also use the word essential to indicate that a property holds a.e. (e.g. “essential bijection”). Theorem 4 (Hahn–Kolmogorov extension theorem): Let X be a set, A0 an algebra of subsets of X, and m : A ! [0, ¥] a s-additive function. If A is the s-algebra generated1 by A , there 0 0 0 exists a measure m : A! [0, ¥] such that m = m0. If m0 is s-finite, the extension is unique. A0 Definition 5 (Cylinder): Let Ak be a s-algebra for k 2 N. Let k1 < k2 < ... < kr be integers 2 A = and Aki ki , i 1, . , r.A cylinder set (also called rectangle) is set of the form [ ] = f g 2 ≤ ≤ Ak1 ,..., Akr xj j2N xki Aki , 1 i r . Definition 6: Let (Xi, Ai, mi), i 2 N, be normalized measure spaces. The product measure space (X, A, m) = ∏i2Z(Xi, Ai, mi) is defined by r X = X [A A ] = (A ) ∏ i and m k1 ,..., kr ∏ mkj kj . i2N j=1 An analogous definition holds if we replace N by Z, i.e. if X consists of bi-infinite sequences. One can see that finite unions of cylinders form an algebra in of subsets of X. By Theorem4 it can be uniquely extended to a measure on A, the smallest s-algebra containing all cylinders. It is often necessary to approximate measurable sets by sets of some sub-class (e.g. an algebra) of the given s-algebra : Theorem 7: Let (X, A, m) be a probability space, and let A0 be an algebra of subsets of X gene- rating A. Then, for each # > 0 and each A 2 A there is some A0 2 A0 such that m(ADA0) < #. Here, EDF := (E n F) [ (F n E) denotes the symmetric difference of E and F. 1.2 The monotone class theorem Definition 8: As sequence of sets fAkg is called increasing (resp. decreasing) if Ak ⊆ Ak+1 (resp. Ak ⊇ Ak+1) for all k. The notation Ak " A (resp. Ak # A) means that fAkg is an increasing (resp. decreasing) sequence S T of sets with k Ak = A (resp. k Ak = A). Definition 9 (Monotone class): Let X be a set. A collection M of subsets of X is a monotone class if whenever Ak 2 M and Ak " A, then A 2 M. Theorem 10 (Monotone Class Theorem): A monotone class which contains an algebra, also contains the s-algebra generated by this algebra. 2 Lebesgue integration Definition 11 (Borel s-algebra / measure): Let X be a topological space. The smallest s-algebra containing all open subsets of X is called the Borel s-algebra . If A is the Borel s-algebra , then a measure m on A is a Borel measure if the measure of any compact set is finite. Definition 12 (Measurable function): Let (X, A, m) and (Y, B, n) be measure spaces, and f : X ! Y a function. We call f measurable, if f −1(B) 2 A whenever B 2 B. Lebesgue integration is concerned with integrals of measurable functions where Y = R (or C) and B is the Borel s-algebra on R. For a detailed construction of the Lebesgue integral we refer to any textbook on measure theory. 1 The s-algebra generated by a collection A0 of subsets of X, also denoted by s(A0), is the smalles s-algebra contai- ning A0, i.e. \ s(A0) = A. A is a s-algebra with A0⊆A Analogously we can define the algebra of subsets of X generated by some collection of subsets of X. We merely note, that a bounded measurable function f can be approximated to arbitrary accu- = n 2 R racy by simple functions of the form fn ∑i=1 licAi , where li , and the Ai are disjoint mea- surable sets. Here, cA denotes the characteristic function of A, i.e. the function with cA(x) = 1 for x 2 A and cA(x) = 0 for x 2/ A. The (Lebesgue) integral of simple functions is given by Z n fn dm := ∑ lim(Ai), i=1 and f is called (Lebesgue) integrable if for any convergent approximations of it by simple func- R tions fn the limit limn!¥ fndm exists and is unique. p p p Definition 13 (L space): For p 2 (0, ¥) the space Lm(X) (sometimes also denoted as L (X, m)) consists of the equivalence classes2 of measurable functions f : X ! C such that R j f jp dm < p R p 1/p ¥ ¥. For p ≥ 1, the L norm is defined by k f kp = ( j f j dm) . The space Lm (X) consists of equivalence classes of essentially bounded functions. ¥ p p If m is finite, then Lm (X) ⊂ Lm(X) for every p > 0. Here is a connection between L functions and continuous functions. Theorem 14: If X is a topological space and m is a Borel measure on X, then the space C0(X, C) p of continuous, complex-valued, compactly supported functions on X is dense in Lm(X) for all p > 0. H¨older’s inequality gives another connection between functions in Lp spaces: if p 2 [1, ¥] and q p q are such that 1/p + 1/q = 1 (with the convention 1/¥ = 0), f 2 Lm(X), and g 2 Lm(X), then one has Z j f gj dm = k f gk1 ≤ k f kpkgkq. For p = 2 the norm k · k2 comes from the inner product Z h f , gi = f g dm, 2 therefore Lm(X) is a Hilbert space. For sequences of functions we have the following results concerning interchangeability of inte- gration and limits. Theorem 15 (Fatou’s lemma): For a sequence f fng of non-negative measurable functions, defi- ne f : X ! [0, ¥] as the a.e. pointwise limit f (x) = lim inf fn(x). n!¥ Then, f is measurable and Z Z f dm ≤ lim inf fn dm . n!¥ Theorem 16 (Lebesgue dominated convergence theorem): Let f : X ! [−¥, ¥], g : X ! [0, ¥] be measurable functions, and fn : X ! [−¥, ¥] be measurable functions such that j fn(x)j ≤ g(x) and fn(x) ! f (x) as n ! ¥ a.e. If g is integrable, then so are f and the fn, furthermore Z Z lim fn dm = f dm . n!¥ Note that non-negative integrable functions define finite measures: Let f : X ! [0, ¥] be integ- rable, then m f : A! [0, ¥], defined via Z Z m f (A) := f dm = f cA dm A is a measure. Here is the converse result: 2Two measurable functions are equivalent if they coincide up to a set of measure zero. Theorem 17 (Radon–Nikodym): Let (X, A, m) be a finite measure space, and n : A! [0, ¥) a second measure with the property3 n(A) = 0 whenever m(A) = 0. Then there exists a non- negative integrable function f : X ! [0, ¥] such that Z n(A) = f dm . A Theorem 18 (Fubini): Let (X, A, m) be the product of the measure spaces (Xi, Ai, mi), i = 1, 2, and let a m-integrable function f : X ! R be given.