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PMATH 451 Course Notes PMATH 451 Course Notes University of Waterloo The One And Only Waterloo 76er Bill Zhuo Free Material & Not For Commercial Use Contents 1 Algebra of Subsets of a Set X ...................................7 1.1 Definitions, Properties, and Some Basic Examples7 1.2 The Trick of the Semi-Algebra9 1.3 Algebra of Sets Generated by An Arbitrary C ⊆ 2X 10 2 Additive Set-Function on An Algebra of Sets .................... 13 2.1 Definition, Properties, Some Basic Examples 13 2.2 An Addendum to the Trick of the Semi-Algebra 15 3 s−Algebras of Sets, Positive Measures ......................... 19 3.1 Definitions, Properties, Some Basic Examples 19 3.2 Continuity Along Increasing and Decreasing Chains 21 4 The Borel s−Algebra, and Lebesgue-Stieltjes Measure .......... 25 4.1 s−Algebra of sets generated by an arbitrary C ⊆ 2X . The notion of Borel s−Algebra 25 4.3 Lebesgue-Stieltjes Measures on BR 27 5 Follow-up on Lebesgue-Stieltjes Measures ..................... 29 5.1 p−systems, and why we have an affirmative answer to Q1? 29 5.2 Caratheodory’s Extension Theorem 30 6 Extension Theorem of Caratheodory ........................... 33 7 Measurable Functions ........................................ 37 7.1 Measurable Functions 37 8 Convergence Properties of Bor(X;R) ........................... 41 8.1 Bor(X;R) is closed under pointwise convergence of sequences 41 8.2 Approximation by Simple Functions 44 + 9 Integration of Functions in Bor (X;R) ........................... 45 9.1 Integral for Simple Non-negative Borel Functions 45 + 9.3 Moving to Bor (X;R) 47 9.4 Discussion around the proof of MCT 49 10 The Space L 1(m) of integrable functions with respect to m ...... 51 10.1 Integrable Function 51 10.4 Equality a.e.-m for functions in Bor(X;R) 53 10.5 Integral on a subset 54 11 Lebesgue’s Dominated Convergence Theorem (LDCT) .......... 57 11.1 The statements(s) of LDCT 57 12 The Spaces L 2(m) ............................................ 61 12.1 What is L p(m) 61 12.2 Equality almost everywhere, and discussion of Lp(m) vs. L p(m) 63 13 L p(m) Completeness ......................................... 65 13.1 Completeness in a seminormed vector space 65 p 13.2 Completeness of (L (m);k·kp) 66 13.4 Modes of Convergence 68 14 The L 2(m) Space and Bounded Linear Functionals ............. 71 14.1 The Inner Product Structure on L 2(m) 71 14.2 Proof of Riesz for L 2(m) 72 15 Inequalities Between Positive Measures ........................ 75 15.1 Linear Combinations and Inequalities for Positive Measures on (X;M ) 75 + 15.2 Integration of Densities from Bor (X;R) \ L 1(m) 76 15.4 An Instance of the Radon-Nikodym Theorem 78 16 Radon-Nikodym Theorem ..................................... 79 16.1 The notion of absolute continuity 79 16.2 The Trick of the Connecting Function 82 5 17 Direct Product of Two Finite Positive Measures .................. 85 17.1 Direct Product of Two Measurable Spaces 85 17.2 Direct Product of Two Finite Positive Measure 86 18 Calculations related to m × n .................................. 89 18.1 Measurability for Slices of Sets and Functions 89 18.3 How to Calculate m × n(E) via Slicing 90 18.7 Theorem of Tonelli 91 19 Sigma-Finite Product Measures ................................ 93 19.1 Direct Product of Sigma-Finite Positive Measures 93 19.2 Sigma-Finite Tonelli Theorem 94 19.3 Fubini Theorem 94 20 Upgrade Radon-Nikodym ..................................... 97 20.1 Upgrade to Sigma-Finite in Radon-Nikodym 97 1. Algebra of Subsets of a Set X 1.1 Definitions, Properties, and Some Basic Examples Definition 1.1.1 — Algebra of Sets. Let X be a non-empty set and let A be a collection of subsets of X. We say that A is an algebra of sets to mea that it satisfies the following conditions: 1. (AS1) /0 2 A 2. (AS2) If A 2 A , then XnA 2 A 3. (AS3) If A;B 2 A , then A \ B 2 A . R 1. The A from Definition 1.1.1 is, in a certain sense, a set of sets. In fact, the first sentence in the definition could be told as: Let X be a non-empty set and let A ⊆ 2X , which is the power set of X. In this course, these calligraphic fonts, like A ;B;M will be used to denote subsets of 2X . 2. The notion of algebra of X sometimes also goes under the name of field of subsets of X. 3. Why we only care about the intersection in (AS3)? This is in fact sufficient for other things we want. Proposition 1.1.1 — Properties of A . Let X be a non-empty set and let A be an algebra of subsets of X. 1. The total set X must be in A 2. If A;B 2 A , then A \ B 2 A 3. If A;B 2 A , then AnB 2 A Tn 4. For all n ≥ 2 and every A1;··· ;An 2 A , we have i=1 Ai 2 A Sn 5. For all n ≥ 2 and every A1;··· ;An 2 A , we have i=1 Ai 2 A . Proof. 1. The combination of conditions (AS1) and (AS2) gives us that X = Xn/0 2 A . 8 Chapter 1. Algebra of Subsets of a Set X 2. We have A;B 2 A =) XnA;XnB 2 A (AS2) =) (XnA) \ (XnB) 2 A (AS3) =) Xn(A [ B) 2 A De Morgan! =) A [ B 2 A (AS2) 3. We can write AnB = A \ (XnB) where A 2 A and where XnB 2 A as well, due to (AS2). Hence, (AS3) assures us that AnB 2 A . 4. Induction on n, where for the base case n = 2 we use (AS3) 5. Induction on n, where for the base case n = 2 we use the statement from part 2. R The moral coming from proposition 1.1.1 is hence this: When dealing with an algebra of sets A , we can do any kind of set operations we want, involving finitely many sets from A , and the result of these operations will still belong to A . Alright, let’s see some concrete examples. X Example 1.1 1. Let X be a non-empty set and let A = 2 . This A obviously satisfies the conditions (AS1), (AS2), (AS3) and is thus an algebra of subsets of X. 2. Let X be an infinite set and let A := fF ⊆ X : F is finiteg [ fG ⊆ X : XnG is finiteg we claim this is an algebra of subsets of X. Let’s check. Let F = fF ⊆ X : F is finiteg and G = fG ⊆ X : XnG is finiteg, so A = F [ G . (a) (AS1): Since the empty set /0 is a particular case of finite set, we have /0 2 F and therefore /0 2 A . (b) (AS2): Let A be a set in A . Then either A 2 F or A 2 G . In both these cases we see that XnA 2 A since ( A 2 F =) XnA 2 G =) XnA 2 A A 2 G =) XnA 2 F =) XnA 2 A Hence (AS2) holds. (c) (AS3) Let A;B 2 A . There are three cases: i. If A;B are finite, then clearly the intersection is finite. So, A \ B 2 A ii. If XnA;XnB are finite, then, (XnA [ XnB) 2 A () Xn(A \ B) 2 A () A \ B 2 A iii. If A 2 F and B 2 G , then, A \ B = (XnA) [ (XnB) 2 A since XnA 2 G and XnB 2 F . Thus, A is an algebra of subsets of X. 1.2 The Trick of the Semi-Algebra 9 3. In this example, consider X = R. Let J be the collection of intervals of R defined as follow J := f/0g [ f(a;b] : a < b 2 Rg [ f(−¥;b] : b 2 Rg [ f(a;¥) : a 2 Rg [ fRg and define ( ) 9k 2 N and J1;··· ;Jk 2 J such that E := E ⊆ R Ji \ Jj = /0for i 6= j and J1 [···[ Jk = E Then, E is an algebra of subsets of R. It is called the algebra of half-open intervals. 4. Non-example: let (X;d) be a metric space, and let T := fG ⊆ XjG is an open setg . This is the notion of a topology from PMATH351. Well, a topology is not necessarily an algebra of subsets of X. Consider the Euclidean metric on R. Then, then complement of an open set is actually closed (and not open), which is not in T . 1.2 The Trick of the Semi-Algebra How do we generate algebra though? Not so clear? Let’s consider the following. Definition 1.2.1 — Semi-Algebra. Let X be a non-empty set and let S a be a collection of subsets of X. We will say that S is a semi-algebra of subsets of X when it satisfies: 1. (Semi-AS1) /0 2 S 2. (Semi-AS2) for every S 2 S , there exists p 2 N and T1;··· ;Tp 2 S such that Ti \Tj = /0 Sp for i 6= j and such that XnS = i=1 Ti 3. (Semi-AS3) whenever S;T 2 S , it follows that S \ T 2 S . R Comparing to the definition of algebra of subsets, we have the first and the third being identical. The only difference is that the second one more relaxed than (AS2). Clearly, (AS2) implies (Semi-AS2). This reminds me of all the half-open rectangles on R2 (a semi-algebra on R2) and all the finite unions of those half-open rectangles give us an algebra of subsets on R2.
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