FUNDAMENTALS OF REAL ANALYSIS by

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1 FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable) fuctios o R, cardiality, Riema itegral, sequeces of real umbers, sequeces of fuctios, poitwise/uiform covergece. OUTLINE OF THE COURSE I. Review of R ad metric spaces II. Lebesgue measure ad measure spaces III. Measurable fuctios ad Lebesgue itegratio IV. Differetiatio ad siged measures V. Product measure spaces VI. Special topics (Lp-spaces, modes of covergece ad Hausdorff dimesio) REFERENCES Measure, Itegratio ad Fuctioal Aalysis, by R. Ash; Academic Press, Liear Operators-I, by N. Duford & J. Schwartz; Wiley-Itersciece, Foudatios of Moder Aalysis, by A. Friedma; Holt, Riehart & Wisto, Real ad Abstract Aalysis, by E. Hewitt & K. Stromberg; Spriger-Verlag, Itroductory Real Aalysis, by A. Kolmogorov & S.V. Fomi; Dover, Real Aalysis, by H.L. Royde & P.M. Fitzpatrick; 4th ed., Pretice Hall, I. REVIEW OF THE REAL NUMBER SYSTEM AND METRIC SPACES I.1. Axiomatic costructio of R. The real umber system is a complete ordered field, i.e., it is a set R which is edowed with additio ad multiplicatio operatios satisfyig the followig axioms 1. (R, +,.) is a field with additive idetity 0 ad multiplicative idetity (R, ) is a partially ordered set compatible with the field axioms i the sese that (i) x, y R, either x y or y x, (ii) if x y, the x + z y + z for all z R, (iii) if x y ad z 0, the xz yz. 3. (Completeess axiom) Every oempty set of real umbers bouded above has a least upper boud (supremum). [Alterative statemet: Every oempty set of real umbers bouded below has a greatest lower boud (ifimum).] Remarks. 1. Ay complete ordered field is isomorphic to (R, +,., ) with Completeess axiom. 1

2 2 2. For a set S R, sup(s) eed ot belog to S. 3. If sup(s) exists, it is uique. (Hece we have set fuctios!) 4. Sice if( S) = sup(s), correspodig statemets also hold for if. 5. R is ubouded (hece so are N, Z ad Q). 6. For ay x, y R, defie maximum ad miimum of x ad y by { x if x y x y = y if otherwise { y if x y ad x y = x if otherwise, respectively. Fact.1 If x + ɛ y for all ɛ > 0, the x y. Proof. Exercise. Exercises. 1. If a < c for all c with c > b, the a b. 2. For A R oempty ad α R a upper boud for A, α = sup(a) if ad oly if ɛ > 0 x A such that α ɛ < x α. 3. If a b a A, the sup(a) b. 4. If A R is bouded (i.e. bouded above ad below) ad B A is oempty, the if(a) if(b) sup(b) sup(a). 5. Let A ad B be oempty subsets of R such that a b a A b B. The (i) sup(a) sup(b) if sup(b) exists, ad (ii) sup(a) if(b). 6. Let A be a oempty subset of R with α = sup(a). If for c 0, ca := {ca : a A}, the cα = sup(ca). 7. Let A ad B be oempty subsets of R ad let C = {a + b : a A, b B}. If sup(a) ad sup(b) exist, the so does sup(c) ad sup(c) = sup(a) + sup(b). 8. Exercises 2, 4, 6-8 are also is true for ifima with obvious (appropriate) chages i the statemets. Fact.2 x R Z + such that > x. Proof. Exercise. Fact.3 (Archimedea Property) If x, y R with x > 0, the Z such that y < x. Proof. Exercise. Corollary. a) If x, y R with x < y, the z Q (Q c ) such that x < z < y. b) x > 0 Z + such that 1 x <. c) ɛ > 0 Z + such that 1 < ɛ. Proof. Exercise.

3 3 Below are some importat iequalities o (fiite) sequeces of real umbers: if a 1, a 2,..., a ad b 1, b 2,..., b are real umbers, the ( a k b k ) 2 ( a 2 k) ( b 2 k) (Cauchy-Schwartz Iequality), ad ( a k + b k 2 ) 1/2 ( a 2 k) 1/2 + ( b 2 k) 1/2 (Mikowski s Iequality). The set R # := R { } with operatios I.2. The Exteded Real Number System. (i) x + = for all x R; x( ) = if x > 0 ad ( x)( ) = ± (ii) + =, ( ) + ( ) = ;.( ) = ; 0.( ) = 0 ad with the order property that < x < for all x R, is called the exteded real umber system. Remarks. 1) is ot defied. 2) sup( ) = is assumed. 3) If a set A of real umbers has o upper boud, the we say sup(a) =. Hece, i R # every set has a supremum (ifimum). I.3. The topology of R. I the ext three sectios, we will provide a short review of some importat cocepts ad facts from real aalysis. For the proofs of most of the statemets oe ca refer to ay oe of the refereces listed above. Of course, its is strogly advised that the reader should try to provide proofs o her/his ow. { x if x 0 For ay x, y R, defie the absolute value of x by x = which ca be x if x < 0, iterpreted as the distace of x to 0. Now, for ay a R the (ope) ball cetered at a with radius r > 0 is the set B(a, r) := {x R : x a < r} = (a r, a + r). For a set A R, a poit a A is said to be a iterior poit if there exists ɛ > 0 such that B(a, ɛ) A. The set of all iterior poits of a set A is called the iterior of A ad is deoted by it(a). A set A R is called ope if every poit of it is a iterior poit. A set A R is called closed if A c is ope. The empty set is assumed ope (ad closed). Fact. 4 a) The uio (itersectio) of ay collectio of ope (closed) sets is ope (closed). b) The uio (itersectio) of ay fiite collectio of closed (ope) sets is closed (ope). Fact. 5 Every oempty ope set of real umbers is a disjoit coutable uio of ope itervals. A real umber p is called a accumulatio poit of a set A R if for every ɛ > 0 we have B(p, ɛ) A. The set of all accumulatio poits of a set A is called the derived set of A ad

4 4 is deoted by A. The smallest closed set cotaiig a set A R is called the closure of A ad is deoted by Ā. Theorem. (Bolzao-Weierstrass) Every bouded ifiite subset of R has a accumulatio poit. Fact. 6 a) For ay A R, it(a) A Ā. b) A set A R is closed iff A = Ā. c) For ay A R, it(a) is the largest ope set cotaied i A. d) Q = R. (Hece, R is separable, i.e. there exists a coutable set A R such that Ā = R.) Exercises. Let A, B R, the: 1) Ā = A A. 2) it(a) it(b) it(a B) ad it(a) it(b) = it(a B). 3) Ā B = A B ad A B Ā B. A collectio U = {U λ : λ Λ} of subsets of R is called a cover for A R if A λ Λ U λ. The collectio U is called a ope cover for A if each U λ is ope. Theorem. (Heie-Borel) Let A R be a closed ad bouded set ad U be a ope cover. The there is a fiite subcollectio of U that covers A. A set C R is called a compact set if every ope cover of it has a subcover cosistig of fiitely may elemets. Corollary TFAE: a) A R is compact. b) A is closed ad bouded. c) Every ifiite subset of A has a accumulatio poit i A. Theorem. (Nested Set Property or Cator Itersectio Theorem) If {F } is a collectio of closed ad bouded set of real umbers such that F F 1 for all 1, the 1 F. I.4. Sequeces of real umbers Recall that a sequece is a fuctio a : N R; for coveiece, we deote sequeces by (a ) 1, where a() = a, 1. By defiitio, a sequece is a ifiite set of real umbers; hece, if bouded, it has at least oe accumulatio poit, say a, by Bolzao-Weierstrass Theorem. If a sequece has oly oe accumulatio poit, it is called the limit of the sequece ad the sequece is called coverget or we say that the sequece coverges to a, ad is deoted by a a or lim a = a. More explicitly, a a if ad oly if ɛ > 0 N Z + such that N a a < ɛ. Fact.7 Let A R. The a A if there exists (a ) A such that a a. A sequece (a ) is called a Cauchy sequece if ɛ > 0 N Z + such that m, N a a m < ɛ.

5 Exercise. If (a ) is a coverget (Cauchy) sequece, the it is bouded (i.e., M > 0 such that a M for all ). Fact.8 a) Every mootoe bouded sequece of real umbers is coverget. b) A sequece of real umbers is coverget if a oly if it is Cauchy. c) (Bolzao-Weierstrass Theorem for sequeces, versio 1) Every bouded sequece of real umbers has a coverget subsequece. Remarks. 1) lim a = meas that c > 0 N > 0 such that N a > c. Similarly, lim a = meas that c < 0 N > 0 such that N a < c. 2) I geeral, if we say a a, the < a < ; if (a ) R #, the a. Observe that a a meas, for ay ɛ > 0, all but fiitely may a s are i the iterval (a ɛ, a + ɛ). A weakeig of this is requirig ifiitely may of a s are i (a ɛ, a + ɛ). Defiitio. A real umber a is called a cluster poit (or accumulatio poit) of a sequece (a ) if ɛ > 0 ifiitely may a (a ɛ, a + ɛ). Equivaletly, a is a cluster poit of (a ) iff ɛ > 0 ad m Z +, m such that a (a ɛ, a + ɛ). Examples. 1) a : 1, 1, 1/2, 1, 1/3, 1, 1/4,.... Accumulatio poits are 0, 1. 2) (( 1) ). Accumulatio poits are -1, 1. 3) ( 1 l ) =2. Accumulatio poit is 0. Remark. A sequece (a ) may have more tha oe accumulatio poits. I that case, it is ot a coverget sequece; but, for each accumulatio poit, it has a subsequece coverget to that accumulatio poit (Exercise). I particular, if a a, the a is a accumulatio poit (the oly oe). Theorem. (Bolzao-Weierstrass Thoerem for sequeces, versio 2) Every bouded sequece of real umbers has a accumulatio poit. 5 Questio. How do we kow that a give sequece has a limit? Fact.9 Every bouded mootoe sequece of real umbers is coverget. Proof. (Sketch) By Bolzao-Weierstrass Thoerem for sequeces, versio 2, the sequece has a accumulatio poit. Sice it s mootoe, it has oly oe accumulatio poit; hece, it must be coverget. Fact.10 (Cauchy Criterio for sequeces) A sequece of real umbers (a ) is coverget if a oly if it is a Cauchy sequece. Questio. Ca we associate a real umber to ay (ot ecessarily coverget) sequece of real umbers? First, recall that supremum ad ifimum of ay bouded set of real umbers exit (if ubouded, they exist i R # ). Now, give ay sequece of real umbers (a ), defie, for k 1,

6 6 a k = if{a k, a k+1, a k+2,... } ad a k = sup{a k, a k+1, a k+2,... }. The, it follows that a k a k+1 ad a k a k+1 for all k 1. Hece, {a k } is mootoe icreasig ad {a k } is mootoe decreasig sequece. Therefore, lim k a k ad lim k a k exist (i R # ). (If (a ) is bouded, the lim k a k ad lim k a k exist i R.) Also, observe that lim k a k = sup k 1 if k {a }, ad lim k a k = if k 1 sup k {a }. Defiitio. For ay sequece of real umbers (a ), defie lim sup a ad lim if a as lim sup a = lima = if lim if sup k 1 k {a }, ad a = lim a = sup if {a }. k 1 k Remarks. 1. For all 1, a k a a k by costructio. 2. For ay i, j 1, we have a i a i+j a i+j a i ; hece, lim if a lim sup a. 3. lim if a = lim sup a if ad oly if (a ) is coverget; i that case, lim if 4. lim if ( a ) = lim sup a. a = lim sup a = lim a. Exercise. Prove that lim if a (lim sup a ) is the smallest (largest) of all the limit poits of the set {a }. I.5. Brief review of real-valued cotiuous fuctios o R. Let A R ad a A. Recall that a fuctio f : A R is cotiuous at a iff ɛ > 0, δ > 0 such that if x B(a, δ) the f(x) B(f(a), ɛ). Fact.11 A fuctio f : A R is cotiuous at a A if ad oly if for ay sequece (a ) A with a a, f(a ) f(a). Fact.12 A fuctio f : A R is cotiuous o A if ad oly if for ay ope set O R, f 1 (O) is (relatively) ope i A. The followig theorems idicate the reaso why we value cotiuous fuctios. Theorem. (Extreme value theorem) Every cotiuous fuctio f : A R, where A R is compact, attais both of its extrema. Theorem. (Itermediate Value Theorem) If f : [a, b] R is cotiuous, where < a < b <, ad f(a) < γ < f(b), the c (a, b) such that f(c) = γ. Recall that a fuctio f : A R is uiformly cotiuous o A if ad oly if ɛ > 0, δ > 0 such that if x y < δ the f(x) f(y) < ɛ for all x, y A. Note that ot every cotiuous fuctio is uiformly cotiuous; however, uder some coditios this is true. Theorem. If f : A R is cotiuous (o A) ad A is compact, the f is uiformly cotiuous (o A).

7 Let A R be a compact set. Defie C(A) = {f : A R : f is cotiuous}. Hece, every f C(A) is uiformly cotiuous. With the usual additio ad scalar multiplicatio of fuctios, C(A) is a vector space over R. Furthermore, if for ay f, g C(A), d(f, g) = sup f(x) g(x), x A the the fuctio d : C(A) C(A) R defies a metric o C(A) (Exercise: Prove this fact); makig it a metric space. Note that d(f, g) is well-defied ad is fiite for ay f, g C(A). Hece, it makes sese to talk about covergece i the metric space C(A). Let (f ) be a sequece i C(A) ad f C(A). Recall that (f ) is said to coverge poitwise to f iff x A, ɛ > 0, N(ɛ, x) such that if N, the f (x) f(x) < ɛ. The sequece (f ) is said to coverge uiformly to f iff ɛ > 0, N(ɛ) such that if N, the f (x) f(x) < ɛ, x A. Clearly, every uiformly coverget sequece is poitwise coverget; but the coverse eed ot hold. However, uder some (rather strog) coditios the coverse holds (such as uder the coditios of Dii s Theorem). Recall that, i the metric space (C(A), d), a sequece (f ) coverges to f C(A) iff ɛ > 0, N(ɛ) such that if N, the d(f, f) < ɛ. It turs out that there is a strog coectio betwee covergece i the metric d ad uiform covergece (o A). Theorem. A sequece (f ) i C(A) coverges to f C(A) iff f f uiformly o A. The metric space C(A) has some other very desirable properties; Fact.13 If f C(A), the M > 0 such that f(x) M for all x A. [Equivaletly, M > 0 such that f B(0, M), where 0 is the zero fuctio.] Theorem. The metric space (C(A), d) is complete. [Hece, every Cauchy sequece of fuctios (f ) C(A) coverges to a fuctio f C(A).] Theorem. (Weierstrass Approximatio Theorem) f C(A) ad ɛ > 0 there exists a polyomial fuctio p ɛ : A R such that d(f, p ɛ ) < ɛ. Corollary. The metric space (C(A), d) is separable. [Hece, there is a coutable dese subset of C(A); amely, the collectio of polyomials o A with ratioal (iteger) coefficiets.] I.6. Axiom of Choice ad its equivalets. Some of the most cotroversial statemets i mathematics are kow as the Axiom of Choice, which is typically assumed as a axiom, ad those statemets proved by usig it ad are equivalet to it. Axiom of Choice. If {X α } α A is a oempty collectio of oempty sets, the Π α A X α is oempty. Let X be a set. Recall that a relatio o X is called a partial orderig if it is reflexive, atisymmetric ad trasitive. If also satisfies the property that the it is called a liear (or total) orderig. if x, y X, the either x y or y x Let (X, ) be a partially ordered set ad E X. A elemet x X is called a upper (lower) boud for E if y x (y x) for all y E. A elemet a X is called a maximal (miimal)elemet of X if x y (y x) the x = y. If every oempty subset E of a partially 7

8 8 ordered set X, ) has a miimal elemet, the X is called a well ordered set ad is called a well orderig. Below are some of the statemets equivalet to the Axiom of Choice. Hausdorff Maximality Priciple. Every partially ordered set has a maximal liearly ordered subset. Zor s Lemma. Let X be a partially ordered set. If every liearly ordered subset of X has a upper boud, the X has a maximal elemet. Well Orderig Priciple. Every oempty set ca be well ordered.