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Measure Theory Measure Theory Andrew Kobin Spring 2016 Contents Contents Contents 0 Introduction 1 0.1 The Discrete Sum . .2 0.2 Metric Spaces . .8 0.3 Partial Orders . .9 1 Measure Theory 11 1.1 σ-Algebras . 12 1.2 Measures . 18 1.3 Borel Measures . 27 1.4 Measurable Functions . 31 2 Integration Theory 37 2.1 Lebesgue Integration . 37 2.2 Properties of Integration . 43 2.3 Types of Convergence . 48 2.4 Product Measures . 52 2.5 Lebesgue Integration on Rn ........................... 54 3 Signed Measures and Differentiation 56 3.1 Signed Measures . 56 3.2 Lebesgue-Radon-Nikodym Theorem . 60 3.3 Complex Measures . 65 3.4 Complex Lebesgue Integration . 68 3.5 Functions of Bounded Variation . 71 4 Function Spaces 76 4.1 Banach Spaces . 76 4.2 Hilbert Spaces . 82 4.3 Lp Spaces . 92 i 0 Introduction 0 Introduction These notes are taken from a graduate real analysis course taught by Dr. Tai Melcher at the University of Virginia in Spring 2016. The companion text for the course is Folland's Real Analysis: Modern Techniques and Their Applications, 2nd ed. The main topics covered are: A review of some concepts in set theory Measure theory Integration theory Differentiation Some functional analysis, including normed linear spaces The theory of Lp spaces By far the most fundamental subject in real analysis is measure theory. In a general sense, measure theory gives us a way of extending the concrete notions of length, area, volume, etc. and also of extending the theory of Riemann integration from calculus. Recall that the Riemann integral of a real-valued function f(x) on an interval [a; b] is defined as the limit of the areas of a sequence of increasingly thinner rectangles that approximate the area under the curve of f: f Z b f(x) dx a If we are to use this limiting process on say a function like the characteristic function ( 1 if x 2 Q χQ(x) = 0 if x 62 Q R 1 then what should the value of 0 χQ(x) dx be? Riemann integration fails to give a value to this integral, but since the \rectangles" summed up by such an integral would all have zero area (Q is disconnected), we should expect the integral to equal 0. We seek to develop an abstract notion of \length" to replace the length of the subintervals in Riemann integration that allows for integration over a broader class of sets. Two important set theoretic constructions are: 1 0.1 The Discrete Sum 0 Introduction 1 Definition. Given a sequence of sets fAngn=1 in a space X, the outer limit of the An is the set of x 2 X such that x 2 An for infinitely many n: 1 1 \ [ lim sup An = An: k=1 n=k This may also be written as the set fx 2 X j x 2 An i.o.g, where i.o. stands for “infinitely often". 1 Definition. Given a sequence of sets fAngn=1 in X, the inner limit of the An is the set of x 2 X such that x 2 An for all but a finite number of n: 1 1 [ \ lim inf An = An: k=1 n=k This is also written fx 2 X j x 2 An e.a.g, where a.a. stands for \eventually always". These may be viewed as generalizations of lim sup and lim inf from real analysis. 0.1 The Discrete Sum A concrete case of integration is found in the notion of a discrete sum: Definition. Let X be a (not necessarily countable) set and let f : X ! [0; 1] be a function. The discrete sum of f over X is defined as ( ) X X f := sup f(x): A ⊂ X is finite : X x2A We will develop the theory of the discrete sum as a case study for what is to come later in the general theory of integration. The main theorems (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem, etc.) will be proven again later with the generalized notion of integration with respect to a measure. Example 0.1.1. If X = N, the discrete sum is just the regular counting sum: 1 N X X X f = f(n) = sup f(n): N N n=1 n=1 One of the most important theorems in the theory of sequences is the Monotone Conver- gence Theorem. This theorem is often proven in an introductory real analysis course. The version stated here is slightly more abstract. Theorem 0.1.2 (Monotone Convergence for Sums). For a sequence of nonnegative functions fn : X ! [0; 1], if fn(x) ≤ fn+1(x) for all x 2 X and for all n 2 N, then X X lim fn(x) = lim fn(x): n!1 n!1 x2X x2X 2 0.1 The Discrete Sum 0 Introduction P Proof. First observe that since fn(x) is an increasing sequence for all x 2 X, fn(x) is P X also an increasing sequence. Therefore limn!1 X fn(x) exists in [0; 1]. Fix n 2 N. By hypothesis, for all x 2 X, f(x) = sup fn(x) ≥ fn(x). Thus for any finite subset A ⊂ X, X X X f(x) ≥ f(x) ≥ fn(x): X A A Taking the sup on the right over all finite A ⊂ X preserves the inequality, so we have X X f(x) ≥ fn(x): X X Since this holds for all n 2 N, taking the limit as n ! 1 again preserves the limit, giving us one of the desired inequalities: X X f(x) ≥ lim fn(x): n!1 X X Now fix a finite set A ⊂ X. Then for all n 2 N, we have X X X fn(x) ≤ fn(x) ≤ lim fn(x): n!1 A X X P (The last inequality uses the fact that X fn(x) is an increasing sequence.) Taking the limit on the left as n ! 1 gives us X X lim fn(x) ≤ lim fn(x): n!1 n!1 A X Finally, taking the sup on the left over all finite A ⊂ X preserves the inequality, producing X X lim fn(x) ≤ lim fn(x): n!1 n!1 X X Since lim fn(x) = f(x), we have proven both inequalities required. P Lemma 0.1.3. If x2X f(x) < 1 then the set fx 2 X j f(x) > 0g is at most countable. Proof. We can write 1 [ 1 fx 2 X j f(x) > 0g = x j f(x) > n : n=1 1 Now each set x j f(x) > n is finite since the discrete sum is assumed to converge. Then the countable union of these finite sets is countable, so the set in question is countable. Theorem 0.1.4 (Tonelli's Theorem for Sums). If X and Y are sets and f : X ×Y ! [0; 1] is a function, then X X X X X f(x; y) = f(x; y) = f(x; y): (x;y)2X×Y x2X y2Y y2Y x2X 3 0.1 The Discrete Sum 0 Introduction Proof. By symmetry, it is sufficient to show the first equality. On one hand, let Λ ⊂ X × Y be a finite subset. Choose finite subsets α ⊂ X and β ⊂ Y such that Λ ⊂ α × β. For any such α; β, we have X X X X X X X X f(x; y) ≤ f(x; y) = f(x; y) ≤ f(x; y) ≤ f(x; y): Λ α×β α β α Y X Y P P P Taking the supremum over all such Λ proves the inequality X×Y f(x; y) ≤ X Y f(x; y). x Going the other way, for each x 2 X choose a sequence of finite subsets βn ⊂ Y such that βx % Y and n X X f(x; y) = lim f(x; y): n!1 x Y y2βn S x If α ⊂ X is finite, then set βn = x2α βn and observe that because βn are finite sets, X X f(x; y) = lim f(x; y) n!1 Y y2βn holds for all x 2 α. Hence X X X X f(x; y) = lim f(x; y) n!1 x2α y2Y x2α y2βn X X = lim f(x; y) since α is finite n!1 x2α y2βn X = lim f(x; y) n!1 α×βn X ≤ f(x; y): X×Y Taking the supremum over all such α ⊂ X, we get the other inequality: P P f(x; y) ≤ P X Y X×Y f(x; y). Therefore equality holds. Theorem 0.1.5 (Fatou's Lemma for Sums). If fn : X ! [0; 1] is a sequence of nonnegative functions then X X lim inf fn ≤ lim inf fn: X X Proof. Define gk = infffn j n ≥ kg so that the sequence (gk) increases from below to lim inf fn. In particular, gk ≤ fn for all n ≥ k, so X X gk ≤ fn for all n ≥ k: X X By the Monotone Convergence Theorem for discrete sums (Theorem 0.1.2), X X X X lim inf fn = lim gk = lim gk ≤ lim inf fn: k!1 k!1 X X X X This proves Fatou's Lemma for discrete sums. 4 0.1 The Discrete Sum 0 Introduction P Remark. If A = X f(x) then for every " > 0, there exists a finite subset α" ⊂ X such that A − " ≤ P f ≤ A. Furthermore, these inequalities hold for any set α containing α .
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