Introduction to Real Analysis (Math 315) - Missouri S&T

1 Introduction to real Analysis ( math 315)Spring 2005 Lecture NotesMartin BohnerVersion from April 20, 2005 Author address:Department of Mathematics and Statistics, University of Missouri Rolla,Rolla, Missouri 65409-0020E-mail 1. The Riemann Stieltjes Functions of Bounded The Total Variation Riemann Stieltjes Sums and Nondecreasing Integrators3 Chapter 2. Sequences and Series of Uniform Properties of the Limit Equicontinuous Families of Weierstra Approximation Theorem7 Chapter 3. Some Special Power Exponential, Logarithmic, and Trigonometric Fourier The Gamma Function12 Chapter 4.

2 The Lebesgue The Lebesgue Measurable Summable Integrable The Signed Measures18iiiivCONTENTSCHAPTER 1 The Riemann Stieltjes Functions of Bounded VariationDefinition ,b Rwitha < b. ApartitionPof [a,b] is a finite set of points{x0,x1,..,xn}witha=x0< x1< .. < xn 1< xn= set of all partitions of [a,b] is denoted byP=P[a,b]. IfP P, then thenormofP={x0,x1,..,xn}is defined by P = sup1 i n xi,where xi=xi xi 1,1 i : [a,b] Rbe a function. We put (P,f) =n k=1|f(xk) f(xk 1)|forP={x0,..,xn} variationoffon [a,b] is defined asb af= supP P (P,f).If baf < , thenfis said to be ofbounded variationon [a,b].

3 We writef BV[a,b].Example nondecreasing on [a,b], thenf BV[a,b].Theorem B[a,b], thenf BV[a,b].Theorem [a,b] B[a,b]. The Total Variation FunctionLemma [a,b] BV[a,x]for allx (a,b).Definition BV[a,b] we define thetotal variation functionvf: [a,b] Rbyvf(x) =x affor allx [a,b].Lemma BV[a,b], thenvfis nondecreasing on[a,b].Lemma BV[a,b], thenvf fis nondecreasing on[a,b].Theorem BV[a,b]ifff=g hwith on[a,b]nondecreasing THE RIEMANN STIELTJES Riemann Stieltjes Sums and IntegralsDefinition ,g: [a,b] Rbe functions. LetP={x0,..,xn} P[a,b] and = ( 1,.., n) such thatxk 1 k xkfor all1 k (P, ,f,g) =n k=1f( k) [g(xk) g(xk 1)]is called aRiemann Stieltjes sumforfwith respect tog.

4 The functionfis calledRiemann Stieltjes integrablewith respect togover [a,b], we writef R(g), if there exists a numberJwith the following property: >0 >0 P P, P < :|S(P, ,f,g) J|< (independent of ). In this case we write bafdg=J,andJis called theRiemann Stieltjes integraloffwith respect togover [a,b]. The functionfis also calledintegrand(function) whilegis calledintegrator(function).Theorem (Fundamental Inequality).Iff B[a,b],g BV[a,b], andf R(g), then bafdg f b ag,where f = supa x b|f(x)|.Example (x) =xfor allx [a,b], thenf R(g) ifff R[a,b].Example C[a,b] andg(x) ={0ifa x tpift < x R[a,b]andg C[a,b].}

5 Thenf R(g)and bafdg= bafg .Example 10xd(x2) = 2 R(g) R(h), thenf R(g+h)and fd(g+h) = fdg+ R(h)andg R(h), thenf+g R(h)and (f+g)dh= fdh+ R(g)and R, then f R(g), f R( g),and ( f)dg= fd( g) = R(g)on[a,b]and ifc (a,b), thenf R(g)on[a,c]. NONDECREASING INTEGRATORS3 Remark , the assumptions of Lemma also implyf R(g) on [c,b].Also, we make the definition abfdg= bafdgifa < R(g)on[a,b]and ifc (a,b), then cafdg+ bcfdg= R(g), theng R(f), and bafdg+ bagdf=f(b)g(b) f(a)g(a).Example 2 1xd|x|= 5 (Main Existence Theorem).Iff C[a,b]andg BV[a,b], thenf R(g). Nondecreasing IntegratorsThroughout this section we letf B[a,b] and be a nondecreasing function on [a,b].

6 Definition P, then we define thelower and upper sumsLandUbyL(P,f, ) =n k=1mk[ (xk) (xk 1)], mk=minxk 1 x xkf(x)andU(P,f, ) =n k=1Mk[ (xk) (xk 1)], Mk=maxxk 1 x xkf(x).We also define thelower and upper Riemann Stieltjes integralsby bafd = supP PL(P,f, )and bafd = infP PU(P,f, ).Lemma bafd , bafd ,P PwithP P, thenL(P,f, ) L(P ,f, )andU(P,f, ) U(P ,f, ).Theorem bafd bafd .Theorem C[a,b]and is nondecreasing on[a,b], thenf R( )on[a,b].41. THE RIEMANN STIELTJES INTEGRALCHAPTER 2 Sequences and Series of Uniform ConvergenceDefinition {fn}n Nbe funtions defined onE R. Suppose{fn}converges forallx E. Thenfdefined byf(x) = limn fn(x)forx Eis called thelimit functionof{fn}.

7 We also say thatfn fpointwise onE. Iffn= nk=1gkfor functionsgk,k N, thenfis also called thesumof the series nk=1gk, write k= (i)fn(x) = 4x+x2/n,x R.(ii)fn(x) =xn,x [0,1].(iii)fn(x) = limm [cos(n! x)]2m,x R.(iv)fn(x) =sin(nx)n,x R.(v)fn(x) = 2n2xif 0 x 12n2n(1 nx) if12n x 1n0 if1n x say that{fn}n NconvergesuniformlyonEto a functionfif >0 N N n N x E:|fn(x) f(x)| .Iffn= nk=1gk, we also say that the series k=1gkconverges uniformly provided{fn}converges (i)fn(x) =xn,x [0,1/2].(ii)fn(x) =xn,x [0,1].Theorem (Cauchy Criterion).The sequence{fn}n Nconverges uniformly onEiff >0 N N m,n N x E:|fn(x) fm(x)| .Theorem (Weierstra M-Test).

8 Suppose{gk}k Nsatisfies|gk(x)| Mk x E k Nand k= k=1gkconverges Properties of the Limit FunctionTheorem funiformly onE. Letxbe a limit point ofEand supposeAn= limt xfn(t)exists for alln SEQUENCES AND SERIES OF FUNCTIONSThen{An}n Nconverges andlimn An= limt xf(t).Theorem continuous onEfor alln Nandfn funiformly onE, thenfis continuous continuous onEfor allk Nand k=1gkconverges unifomlyonE, then k=1gkis continuous a metric space. By C(X) we denote the space of all complex-valued, continuous, and bounded functions onX. Thesupnormoff C(X) is definedby f = supx X|f(x)|forf C(X).Theorem (C(X), d(f,g) = f g )is a complete metric be nondecreasing on[a,b].

9 Supposefn R( )for alln Nandfn funiformly on[a,b]. Thenf R( )on[a,b]and bafd = limn bafn(x)d .Corollary be nondecreasing on[a,b]. Supposegk R( )on[a,b]for allk Nand k=1gkconverges uniformly on[a,b]. Then ba k=1gkd = k=1 bagkd .Theorem differentiable functions on[a,b]for alln Nsuch that{fn(x0)}n Nconverges for somex0 [a,b]. If{f n}n Nconverges uniformly on[a,b], then{fn}n Nconverges uniformly on[a,b], say tof, andf (x) = limn f n(x)for allx [a,b].Corollary differentiable on[a,b]for allk Nand k=1g kisuniformly convergent on[a,b]. If k=1gk(x0)converges for some pointx0 [a,b], then k=1gkis uniformly convergent on[a,b], and( k=1gk) = k=1g Equicontinuous Families of FunctionsExample (x) =x2x2+(1 nx)2, 0 x 1,n familyFof functions defined onEis said to beequicontinuousonEif >0 >0 ( x,y E:|x y|< ) f F:|f(x) f(y)|<.

10 Theorem compact,fn C(K)for alln N, and{fn}n Nconvergesuniformly onK. Then the familyF={fn:n N}is sequence{fn}n Nis calledpointwise boundedif there exists a function such that|fn(x)|< (x) for alln N. It is calleduniformly boundedif there exists anumberMsuch that fn Mfor alln WEIERSTRASS APPROXIMATION THEOREM7 Theorem pointwise bounded sequence{fn}n Non a countable setEhas a subse-quence{fnk}k Nsuch that{fnk(x)}k Nconverges for allx (Arzel`a Ascoli).SupposeKis compact and{fn}n N C(K)is point-wise bounded and equicontinuous. Then{fn}is uniformly bounded onKand contains asubsequence which is uniformly convergent Weierstra Approximation TheoremTheorem (Weierstra Approximation Theorem).