MEASURE THEORY T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Introduction: Riemann vs Lebesgue 2 2. Small subsets of Rd 2 3. About functions behaving nicely outside a small set 9 4. σ-algebras and measurable spaces 17 5. Measures and outer measures 22 6. Lebesgue measure on Rd 24 7. Pre-measure ! outer measure ! measure 30 8. Measurable functions 36 9. Approximating measurable functions with simple functions 40 10. Limit theorems for measurable functions 42 11. Integration with respect to a measure 46 12. Properties of the integral, and limit theorems 48 13. Lp spaces 53 14. Product spaces and product measures 56 15. Fundamental Theorem of Calculus (FTC) 59 16. Measures on metric spaces 61 17. Measure Theory and Functional Analysis 65 Prerequisites: Real Analysis and basic Metric Space Theory. Convention: Axiom of choice and continuum hypothesis will be assumed throughout. Topics for self-reading: Banach-Tarski paradox, Littlewood's three principles. Our approach: We will do measure theory first, and then will do integration theory. We will also discuss the interaction of measure theory with topology in Rd. Some textbooks for further reading: 1. R.G. Bartle, The elements of integration and Lebesgue measure, 1995. 2. D.L. Cohn, Measure Theory, 2nd Edn., 2013. 1 2 T.K.SUBRAHMONIAN MOOTHATHU 3. H.L. Royden, Real Analysis, 3rd Edn., 1988. 4. E.M. Stein and R. Shakarchi, Real Analysis (Princeton Lectures in Analysis, 3), 2005. 1. Introduction: Riemann vs Lebesgue Measure theory helps us to assign numbers to certain sets and functions - to a measurable set we may assign its measure, and to an integrable function we may assign the value of its integral. Lebesgue integration theory is a generalization and completion of Riemann integration theory. In Lebesgue's theory, we can assign numbers to more sets and more functions than what is possible in Riemann's theory. If we are asked to distinguish between Riemann integration theory and Lebesgue integration theory by pointing out an essential feature, the answer is perhaps the following. Riemann integration theory 7! finiteness. Lebesgue integration theory 7! countable infiniteness. Riemann integration theory is developed through approximations of a finite nature (eg: one tries to approximate the area of a bounded subset of R2 by the sum of the areas of finitely many rectangles), and this theory works well with respect to finite operations - if we can assign numbers to finitely many sets A1;:::;An and finitely many functions f1; : : : ; fn, then we can assign numbers to A1 [···[An, f1 +···+fn, maxff1; : : : ; fng, etc. The disadvantage of Riemann integration theory is that it does not behave well with respect to operations of a countably infinite nature - there may not S P 1 1 f 2 Ng be any consistent way to assign numbers to n=1 An, n=1 fn, limn!1 fn, sup fn : n , etc. even if we can assign numbers to the sets A1;A2;:::; and functions f1; f2 :::. Lebesgue integration theory rectifies this disadvantage to a large extent. In Riemann integration theory, we proceed by considering a partition of the domain of a function, where as in Lebesgue integration theory, we proceed by considering a partition of the range of the function - this is observed as another difference. Moreover, while Riemann's theory is restricted to the Euclidean space, the ideas involved in Lebesgue's theory are applicable to more general spaces, yielding an abstract measure theory. This abstract measure theory intersects with many branches of mathematics and is very useful. There is even a philosophy that measures are easier to deal with than sets. 2. Small subsets of Rd It is possible to think about many mathematical notions expressing in some sense the idea that a subset Y ⊂ Rd is a small set (or a big set) with respect to Rd. We will discuss this a little as a warm-up. We will also use this opportunity to introduce Lebesgue outer measure. Suppose you have a certain notion of smallness or bigness for a subset of Rd. Then there are some natural questions. Two sample questions are: MEASURE THEORY 3 1. If Y ⊂ Rd is big, is Rd n Y small? S ⊂ Rd 1 2. If Y1;Y2;::: are small, is n=1 Yn small? For instance, consider the following two elementary notions. Saying that Y ⊂ Rd is unbounded is one way of saying Y is big, and saying that Y ⊂ Rd is a finite set is one way of saying Y is small. Note that the complement of an unbounded set can also be unbounded and a countable union of finite sets need not be finite. So here we have negative answers to the above two questions. Exercise-1: Find an uncountable collection fYα : α 2 Ig of subsets of R such that Yα's are pairwise disjoint, and each Yα is bounded neither above nor below. To discuss some other notions of smallness, we introduce a few definitions. Definitions: (i) We say Y ⊂ Rd is a discrete subset of Rd if for each y 2 Y , there is an open set U ⊂ Rd such that U \ Y = fyg. For example, f1=n : n 2 Ng is a discrete subset of R and Zd is a discrete subset of Rd. (ii) A subset Y ⊂ Rd is nowhere dense in Rd if int[Y ] = ;, or equivalently if for any nonempty open set U ⊂ Rd, there is a nonempty open set V ⊂ U such that V \ Y = ;. If f : R ! R is a continuous map, then its graph G(f) := f(x; f(x)) : x 2 Rg is nowhere dense in R2 (* G(f) is closed and does not contain any open disc). Also, fA 2 Rd×d : det(A) = 0g is nowhere dense in Rd×d. (iii) A subset Y ⊂ Rd is of first category in Rd if Y can be written as a countable union of nowhere dense subsets of Rd; otherwise, Y is said to be of second category in Rd. For example, Y = Q × R S R2 is of first category in since Y can be written as the countable union Y = r2Q Yr, where 2 Yr := frg × R is nowhere dense in R . (iv) (The following definition can be extended by considering ordinal numbers, but we consider only non-negative integers). For Y ⊂ Rd and integer n ≥ 0, define the nth derived set of Y inductively as Y (0) = Y , Y (n+1) = flimit points of Y (n) in Rdg. We say Y ⊂ Rd has derived length n if Y (n) =6 ; and Y (n+1) = ;; and we say Y has infinite derived length if Y (n) =6 ; for every integer n ≥ 0. For example, Q has infinite derived length (since Q = R), and f(1=m; 1=n): m; n 2 Ng has derived length 2. Q d d (v) We say A ⊂ R is a d-box if A = Ij, where Ij's are bounded intervals. The d-dimensional Q j=1 d j j × × − volume of a d-box A is V old(A) = j=1 Ij . For example, V ol3([1; 4) [0; 1=2] ( 1; 3]) = 6. Exercise-2: If U ⊂ Rd is a nonempty open set, then U can be written as a countable union of pairwise disjoint d-boxes (these d-boxes may not be open). [Hint: Let B be the collection of all open d-boxes whose vertices have rational coordinates. For each x 2 U, there is B 2 B with 4 T.K.SUBRAHMONIAN MOOTHATHU S x 2 B ⊂ U. Hence U can be written as a countable union U = Bn with Bn 2 B. Now note that S n n n Bn+1 j=1 Bj can be written as a finite union of pairwise disjoint d-boxes.] (vi) The d-dimensional Jordan outer content µ∗ [Y ] of a bounded subset Y ⊂ Rd is defined as P J;d S ∗ f k 2 N ⊂ k g µJ;d[Y ] = inf n=1 V old(An): k ; and An's are d-boxes with Y n=1 An . (vii) The d-dimensional Lebesgue outer measure µ∗ [Y ] of an arbitrary set Y ⊂ Rd is defined as P L;d S ∗ f 1 ⊂ 1 g µL;d[Y ] = inf n=1 V old(An): An's are d-boxes with Y n=1 An . ∗ ≤ ∗ ⊂ Rd ∗ ∗ [101] We have that µL;d[Y ] µJ;d[Y ] for any bounded set Y , and µL;d[A] = µJ;d[A] = V old(A) for any d-box A ⊂ Rd. S S k 1 Proof. Any finite union n=1 An of d-boxes can be extended to an infinite union n=1 An of d-boxes without changing the total volume by taking An's to be singletons for n > k. This observation ∗ ≤ ∗ ∗ yields that µL;d[Y ] µJ;d[Y ]. It is easy to see µJ;d[A] = V old(A) if A is a d-box. It remains ∗ to show µ [A] ≥ V old(A) when A is a d-box. First suppose A is closed. Then A is compact L;d S d 1 by Heine-Borel. Let " > 0 and let A1;A2;::: ⊂ R be d-boxes such that A ⊂ An and P n=1 1 ∗ 2 N ⊂ n=1 V old(An) < µL;d[A] + ". For each n , let Bn be an open d-box with An Bn and n V old(Bn) < V old(An)+"=2 . Then fBn : n 2 Ng is an open cover for the compact set A. Extracting P P ≤ k ≤ 1 n ∗ a finite subcover, we have V old(A) n=1 V old(Bn) n=1(V old(An) + "=2 ) < µL;d[A] + 2".