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Measure Theory and Integration Michael E. Taylor Graduate Studies in Mathematics Volume 76 Editorial Board David Cox Walter Craig Nikolai h anov Steven G Krantz David Salt man (Chair) 2000 MathfJnatu;s S1tblH t Cla istfication Primary 28 -01 For additional inform,ltion and updates on this book, visit www.ams.orgjbookpagesjgsm-76 Library of Congress Cataloging-in-Publication Data Tay lor, i\Iichael Eugene 1946 Measure theory and integration / i\Iichael E Taylor p em -(Graduat() studiE s in mathpmatics. ISSN 106-7 119 . \ 76) Includes bihliographical referencE s ISBN-13 978-0-8218-4180-8 1 ME asure theory 2 Riemann integrals 3 ConvergE n( ( Probabilit iE S II Series QA312 T 387 2006 515 12 dc22 TitlE 20060456.f) Copying and reprinting. Individual reitder~ of this publi( ,ttion and nonprofit Iihr,lri( s a( ting for thpm are pt rmitted to make fair USE of ttl( matE rial Such ,.s to copy a chapt(,J for use in teaching 01 research pprmis~ion is grantpd to quotp brief pas~ages from this publication in reviews. provided the customary acknowledgment of the source is given Republication systematic copying OJ multiple reproduction of any material in this publication is permitted only under licensE from the Ameri( all I\IathE matical SociE ty Reque;,ts for SUi h pprmi~qion should he addres~(·d to the Acquiures and Complex l\:leaed on a one-semester course on measure theory, which I have taught several times. The prerequisite for the course is an intro- ductory analysis course, covering such matters as metric spaces, uniform convergence of functions, the contraction mapping principle, and aspects of multi-variable calculus, including the inverse function theorem. For the convenience of the reader, some of this material is briefly treated in some appendices. The core topic f0r the course treated here is the theory of measure and integration, associated especially with the work of H. Lebesgue, though of course many other mathematicians have contributed to this central subject. We mention particularly E. Borel, 1\1. Riesz, J. Radon, 1\ 1. Frechet, G. Fubini, C. Caratheodory, F Hausdorff, and A. Besicovitch. among the classical founders. \Ve begin with an introductory chapter on the Riemann integral, for functions defined on an interval [(1, b] in JR We devdop some of tll( proper ties of thc Riemann integral, including a proof of the Fundamental Theorem of Calculus. \Ve see that while continuouf functiont are Riemann integrable, fome very reasonable-looking functiont are not. In Chapter 2 we construct Lebesgue measure on JR. We emphasize that the key difference between Lebesgue meature of S c JR and the “content“ of S, arising from the Riemann integraL is that the content is approximated by taking finite coverings of S by intervab, while the Lebesgue measure is approximated by taking countable (perhaps infinite) cover ings of S by intervals. In Chapter 3 we define the Lebctgue integral and establish tome basiC properties, such at the Monotone Convergence Theorem and the Lebesgue - VII VUl Introduction Dominated Convergence Theorem We integrate mea::mrable function:“ de- fined on general measure :“paceb. Though at t hi~ point we have only con- ::;tructed LebeEgue mea::;ure on JR, the ha::;ic theon“ of integration i:“ not mOle complicated on general meabure bpace~, and purbuing it help:“ dar ify what orlt :“hould do to conEtruct more general llW(l::;UlC i III Chapter -1 we in- troduce LP bpaces, con~i~ting of measurable funetiollb f buch that I tiP i:“ integrable or, more precibdy. of equivalence cla::;bPb of ::;11ch function::;, \vherC we ba:v fr “- h provided thet e functionb differ only on a ::;et of mea~ure / ero. If f1 i::; c1 meabure on a space X. we btudv LP(X, f.1) ab a Banach bpace for 1 ::; p ::; ex, and in particular we ~tudv L2(X, f1) a::; a Hilt)( rt :“pacC . We develop borne Hilbert :“pacp tlworv aud apply it to Pbtablibh the Radon- Nikodym Theorem. comparing two mea::;ures f.1 and 1/ whcn 1/ ib “ab:“olutcl:v continuou::;“ \vith respcct to f.1 COllbtructing meabure::; other than Lebc::;gup llleasUH on JR i::; an impor- tant part of measure theory, and we begin thi::; ta:“k in Chapter 5. giving borne woeful general methods, e::;pecially elm to Caratheodorv, for making ::;uch construction:“, establishing that tllPY arC mectsureb, and idelltif~ ing cer- tain types of sets as llwasurable Thc firEt concrete applicatioll of thib is made in Chaptcr 6, in the conbtruction of thC “product Illeabure on X x y, when X has a meabure f.1 and Y hab the meabure v. The integral with re- spect to the product lllcasure I) x v ib comparcd to “iterated integrab: in theoremE of Fubini and TOllelli In Chaptn 7 \ve (onstruct Ld)ebgue measure on JRfl for n 1, as a product meabure. Ye btudy how the Lebpsgue illtegrdl Oll JRn tra,ntoforrn~ under an invertiblc linear transformation on JRn and, mOlt gencrally. under a C1 diffeornorphibm. \Ve go a bit furthe r, considering trdn:“formatioll via a Lipbchitz hOllH omorphibIrL “nd we establibh a, leblllt uudC r the hypotlH - si::; that the transfor mation is diffncntiable cllmo::;t evn:vw her e. a proper ty that will be studied further in Chapter 11. We extend the tocope of thc n- dimensional integral in another dire( tion, cou::;tr uet ing b urfacT measure on an n-dimcllsional surface 1H in JRm. Thb is dOlW in terms of the Riemann metric tenbor induced on lo.J. \\Te go furthpI (1nd ditoCUbb integration on mOl c general Riemannian manifolds Thib central chapter containb a lc1rgel num- [wr of exerciscto than the other b. dividpd into beveral bets of exercibeb. After the first toet. of a nature par dUd to exercise set::; for other chapter~. thcn ib a sC t relating the Riemann and Lpbebgue integrab on JRn, pxtending the pre- viOll~ dbcu%ioll of the rclationbhip between the material of Chaptpr 1 and that of Chapter ; 2 3, an exercise :,;et on determinant~, and an exercise set on row reduction on matrix prodllCtb, providing linear algC bra background for thp proof of the change of variable formula There i i also an exercibl bet Introduction IX on the conncctivity of Gl (n, JR) and a set 011 integration on certain matrix groupf-l In Chapter 8 \vC di~cuf-lf-l “signed llleitSUrEf-l,“ which dificr from thc mca- sures conf-lidered up to th,tt poillt onlv in that they can take negativc as well ,1: pOf-litive values The kC v rc~ult established tlwrc i:, thc Hahn de- composition, 1/ = 1/+ -1/-for it ~igned meaS11lC 1/, on X, where 1/+ and //-arc positive measurC :“ with di:,joint f-lupport:, This allows u~ to cxtend the Radon-Kikoclvrn Theorem to the caf-le of a signed measure 1/. absolutely continu()u:, with rC ~pect to a (positive) mea:,ure 11 \Ve con:,ider a further extent:ion. to complex meat:UIes, though this b completelY routine, In ChaptcI 9 \ve again take up the study of LP space:, and pur:,ue it a bit further. \Ye identify the dual of the Bandch :-,pace LP(X. !1) with Lq(X, /1), where 1 :::; p UI( S a:, a tool in the delllonstration, \Ve study some integral operator: induding convolutioll operatort:, amongst others, anel derive operator bounds on LP :,pacef-, \Ve cont:ider the Fouricr transform :F and prove that :F is paces. Hk P(JRrt) conf-listing of functions whose derivatives of order :::; k defin( d in a :,uitable weak sen:,t“ belong to LP(JRrt) Certain Sobokv ~pacC :-, are shown to ( on~ist C ntirply of bounded cOIltinuOUf- functions, evpn HOlder cOlltinuous. on JRrt, Thif-l subject is of grt at u oe in the study of partial diffeH ntial eCjuations, though :,uch applicatiollf- are not made here, The most significant application we In,lke of SobolE v sIMce theory in thef-e note:, appears ill the followillg chapter, \Ve mention t hdt in Chapter 10 u r 1 ain rc:-,ults on the v,, pak df rivatiw dcppncl on thE Fundamental Th( orem of Calculu:-, for the Riemann intPgraL it is for this rea~on that we included a proof of t his result in Chapter l. Chaptel 11 deab with variou:-, results in the arpa of ahno:,t-everywhert cOllvergencC , The ba:-,ic rt“mlt. called Lebesgue:-, differentiation t heorpm, is that a functioll f E J) (JRrl) i:, e(jlMl for .=tlmof-lt .=tll .r, E JRn to the limit of it:-, aver age:, over balb of radiu:, r, centered at J, as r --+ 0, LJ nder a slightly :,trollger (ondition on .1 , we :-,ay .1 is ct Lpbe:,glH. point for .f, more generally thert i:-, the notion of dll LP-Lc])( sglle point, and one :-,hows that if j E LP (JRII), then almost (,VCIY r E JRH i:-, dn LP -Lebe:,gue point for f, provided 1 :::; p TI (01 p = n = 1), then .f i8 differentidbk almo“t evelywhere: in fact. f b differentiable at every LP-Lelw,)gue point of the weak derivative Vf ,,\lith thi“ re8ult we um complete the delllOll8tratioll of thc lC 8Ult froUl Chdptel 7 thd,t the Chd,llge of variahle fornmLl for tIl( intpgral cxtend o to Lip8chitz homeomorphislll,) ;“Iaking u,)p of Rademachn “ Them CIll. wc 8110\\ that a Lip::,chitz function can bp etlteH d off a 81I1all )et to ,“iPld et ( 1 function Thc“p results alC impoltant ill the study of Lip8chit6 ourfacc:-, in JR“ The covering lelllma WC U8(, fOl t hc re,)lllt8 of Chaptel 11 mentioned abovp is \Veiner 8 Coveling Lelllllla \Ve abo di8cu o o COWl iug ICllllll(\8 of Vit ali etnel of Besicovitch and show how ll( “i( ovitdl o rl )ult kad8 to Pxtplhions of t hc Lebesgue diffel cntiation t heOl ( m In Chaptel 12 we con:“tl1lct 1 -dimeu )ionai Hc\u )dorff nW,l,,111C H . on a separablc metlic “pace X. for dnv r E [0. )C) In C(-I8e S i“ a Bm el set in JRI!. one has Hn(s) = ￡11 un, Lebe:“guc llletlSme thi ) i:-, lloi a stlelightfor\\“cud consequence of the dcfinition (unles,) TI = 1) alld ib proof leqnirc“ SOUle effort. In parti(111d,r. the plOof require “ a cO l, Clillg lemma \Ye C xtend this analysis to show that n-dimensional HausdOlff mphnl ill Chaptel 11 an invaluable he ll These plO\“idp l)dsic ll oulh in W ometIic nll d:- llIe theory Therp b a gH at deal mOlP to g A good o\ PlVicw C. SOlllt of innedibk hpauty (,Cllh d “fIdl tell o, for which H is gennclllE for c\ vaJu( r rt Z \Ve tonch t hi ) ouly hI iefi,“ one ( an COlhUlt [Ed]. [Mdb]. [Fal] and [PRJ tor mOll material ou ira( t,tb. In Chapter 1:3 \\ l how how a pO oitiVl liuear functional on C(X). the “Pd.C c of cOlltinuou“ functioll o on X. givP ) r i oc to [t (pofitive) lllPac,Ul( on X. when X if (1 compact lllPtric )pa(e, and how a bounded lincar hmctiOlMI on C(X) gives ri t, to a igned 1ll( a oUl( on X. Out of thi ) com( compactm ,,:“ rpsult~ for bounded 8ch of lllea iUrp o Thp op H “ult:-. exteud to tIl( (d,)P where X if: a genpr al compact HausdOl H “pace tlPatrnenh of thi ) (all ht, found in :“( VCT (11 plan :-,. including [Fol] and [Ru] Thc argumcut i “ )oDlewhat “impll I Introduction Xl when X i8 metrizablc; in particular, we can appeal to re8ult~ from Chapter 5 for a lot of the technical work ChaptC r~ 14-17 explore connections between mea:6 TJ f(.r). l\1ean ergodic the- orern8 and Birkhoff“s Ergodic Theorem treat ￡P-norrn behavior and point- wi8e a e behavior of Akf(.r), tending to a limit Pf(x) as k ----+ 00 Ergodic transformations are those for which such limits are conv distributions In Chapter 16, we construct \Viener meaf,ure on the set of continuous paths in ]Rn, describing the probabilistic behavior of a particle undergoing Brownian motion \Ve begin with a probability measure TV on a countable product 13 of ( olllpactincations of ]Rn, first defining a po~itive linear func- tional on C( 13) and tllPn getting the lIlea~Ule via the re8ult~ of Chapter 13. The index set fOI the (“ounta,ble product i~ Q+. the ~et of rational numbers ~ O. The spc1ce of eontinuous paths i~ naturally identified with a subset 130, 8hoWll to bf:: uggestions for improvements. During the course of writing this book, my H seardl ha:“ been supported by ~SF grants, including most recently NSF Grant #0456861 11ichael E. Taylor Chapter 1 The Riemann Integral The Riemann integral is a fundamental part of calculus and an essential precursor to the Lebesgue integral. In this chapter we define the Riemann integral of a bounded function on an interval I = [a, bj on the real line. To do this, we partition I into smaller intervals. A partition P of I is a finite collection of subintervals {Jk : 0 ~ k ~ N}, disjoint except for their endpoints, whose union is I. We can order the Jk so that Jk = [Xk Xk+l], where (1.1) Xo -Q, if P is formed by partitioning each interval in Q. Equivalently, P -Q if and only if all the endpoints of Q are also endpoints -1 2 1 The Riemann Integ! al of P It b f asy to ::,ee that any two partition::, have a common refinement: ju::,t take the union of their endpoint::, to form a Hew partition Note abo that Consequentlv. if PJ are any hvo partition::, and Q is a commOll refinement. we have (1 5) ~ow. whenever f . 1 ----* lR i“ bounded, the following quantitie::, are \vell defined (1 6) 1(1) = iuf 1p(f). PElI(J) I(f) = ::,up L,.,( f). PED(J) where II(I) if the set of all partitiolls of I CIE arly, hy (1 5), I(f) :; 1U) We then say that f i::, Rl,errwnn mtegrable provided 1(f) = IU), and in such a ca::,e, we set (1.7) .l f(x) dx = 1(f) = I(f). 1 \VE will denotE the ::,et of RiE lll(1ull integrable functiou::, on I by R(I). \Ve deri\ ( some basic plOpertie:: of the Riemann integral Proposition 1.1. If f, q E R(I). then f + 9 E R(Ji. and (1 tl) /(f + q) d.r = .I f (iJ; + .I 9 dx. I f I Proof. If ,J“. i::, any ,-;ubinter val of I, then “up (f + g) :; sup f + ::,up g, J, h h so, for any pcutition P, we have 1p(f + g) :; 1p(f) + 1p(g) Also, using common refinemt nb, we can simv uanfO U5ly applOximatc 1(f) and 1(g) by 1,., (f) and 1 p (g) Thus the chdn1cterizdtion (1.6) implies 1 (f + g) :; 1 (f) + 1 (g). A par allel argument implies IU + (}) 2: IU) + 1(g), and the propo::,ition follow::, Next, th( I{ is a fair supply of Hit mann integrable funetions 1. The Riemann Integral 3 Proposition 1.2. If f is continuous on I, then f is Rwmann integrable. Proof. Any continuous function on a compact interval is uniformly contin- uou~; let w( 0) be a modulu~ of continuity for f, 50 (19) Ix -YI :S 0 ===? If(x) - f(y)1 :S w(15). w(o) ---+ 0 as b ---+ O. Then (1.10) maxsize (P) :S 0 ===? Jp(J) -Ip(J) :S w(b) . ￡(1), which yield~ the proposition. Thi~ argument, showing that evelY continuous function is Riemann in- tegrable, abo provider :S Ipo (J) + E. I AI = sup If(￡)I. 0 = minsize(Po)· I Proposition 1.3. Under the hvpothe~e8 above, if P zs any partitwn of I satisfvmq (1 13) then (1.14) -J 2Af Jp(J) -El :S. f d.r :S Ip(J) + El· wdh El = E + T￡(I)· J Proof. Consider on the one hand tho~E intE rvals in P that are containec in intervals in Po. and on the other hand those intervab in P that are no contained in intE rvab in Po (who~e lE ngth~ ~um to :S f(1)/k). Let PI be tll( minimal common refinement of P and Po. \Ve obtain The following corollary is sometimes called Darboux s Theorem. 4 1. The Riemann Integral Corollary 1.4. Let Pv be an lj sequence of partitions of I into v intervals Jvb 1::::: k ::::: v, such that and let C,vk be any choice of onf pond HI each mterval Jvk of the partitwn PI/ Then. whrneucr f E R(1), (1.15 ) v / f(r) dx = }~~ L f(~l/k) f(Jvk) [ k=l The S11m on the right side of (1.15) is called a Riemann sum. One should be \varned that, Ollce such a specific choice of Pv and ~vk has been made. the limit on the right side of (1.15) might exist for a bounded function f that is not Riemann integrable This and other phenomena arc illustrated by the following example of a function which is not Riemann integrable For :t E 1, set (116) 1)(.1 ) = 1 if x E Q, 17(,1:) = 0 if x tI-Q, where Q is the t.et of ratwnal numbert. Now ever:v interval J c I of positive length containt. pointt. in Q and points not in Q, t.o for any partition P of I we haw /p(19) = f(J) and Ip(19) = 0, and hence (1 17) /(1 )) = f(1), I( J) = O. Note thclt, if Pv i~ a partition of I into v equal flubintervals, thell we could pick each ~lJh to be rational, in which case the limit on the right side of (1 15) would be f(1). or we could pick each ~vk to be irrational, in which case thifl limit would be zeIO. Alternatively, we could pick half of them to be rational and half to be irrationaL and the limit would be €(I) /2 Let .fA E: R(l) be a uniformly bounded, monotonically increasing t.e- quen(( of functioll~ TheIl there is a bounded function f on I such that, as k ----7 x“ (1 18) fk(X) / j (x), Vx E I It would be det.irable to conclude that f is integrable and (1.19) / h,(x) d“ ----7 J f(x) dx. I I 1 The Riemann Integral 5 A shortcoming of the Riemann integral is that such a limit might not belong to R( 1). For example, since I n Q is countable, let I n Q = {cr, C2, C3, }, and let (1 20) I9k(X) = 1 if x E {cJ . ,Ck}, 0 otherwise. It is easy to see that I(79 k) = J( I9t.,) = 0, so each 19k E R(I). But, as k - 00, (1 21) defined by (1.16), which is not in R(I) The Lebesgue theory of integra- tion remedies this defect If fk are Lebesgue integrable, and if one uses in (1 19) the Lebe~gue integral (which coincides with the Riemann integral for function~ in R(I)), then (1.18) =:::;. (119) This is known as the Mono- tone Convergence Theorem, and it will be seen to be a central result in the Lebesgue theory. Associated to the Riemann integral is a notion of ~ize of a set S, called content. If S is a subset of I, define the “characteristic function“ (1.22) xs(X) = 1 if xES, 0 if x rt s. We define “upper content“ cont+ and “lower content“ cont-by (1.23) We say S “has content,“ or “b contented“ if these quantitie~ are equal, which happens if and only if XS E R(I). in which case the common value of cont+ (S) and cont -(S) is (1 24) m(S) = J xs(.r) dx T I t i~ easy to see that (1.25) N cont+(S) = inf{I::f(,h): S C ,f} U . U IN}, k=1 where ,h are intervals. Here, we require S to be in the union of a jmit, collection of intervals. The key to the comMuction of Lebesgue mea~ure b to cover a set S b~ a countable (either finite or mfinite) ~et of intervals. The outer measure 0 ScI will be defined by (1 26) m*(S) = inf{I::f(J k) . S c U ,ft.,} k2: 1 k2:l 6 1. The Riemann Integral Here {Jd is a finite or coulltably infinite collection of intervals. Clearly (1.27) m*(S) S cont+(S). Note that. if S = InQ. thm xs = 19. definC d bv (1.16) In this case it i::, easy to see that C Ont+(S) = tel). but m*(S) = O. Zero is the right“ lll( asure of this set \Ve develop a few morc properties of tIlt Riemdnu integral. It is useful to note that il f d.r: is additive in I. in the following sense Proposition 1.5. If a JR. !l = fl [0.1? h = rl [b rl then ( 1.28) 1 E n([a, eJ) ~ 11 E n([a. bJ) and 12 E nOb. c]). and. ~r thzs holds. (1.29) /“ ib [( 1 dx = 11 d.1~ + h d:r: . . a ? a ? Ii Proof. Since any partition of [a. r] has a refinement for which b is an end- point. we may as well consider a partition P = PI U P2. where PI is a partition of [a. bJ and P2 it; a pcutition of [b, f j. Then (130) so (1 :31) Since both terms ill braces in (131) are;::: O. w( have equivalence in (1.28) Then (1 29) follows from (1.:30) upon taking sufhciently fine pclrtitions. Let I = [a. bJ If 1 E n(I). then f E n([a r]) for 0, (1.37) 1 1 fT+h h [g(:1: + h) -g(x)] = h .Ix f(t) dt. If f b continuou~ at x, then, for any E O. there exi~ts 8 ° such that If(t) - f(:r)1 :S E whenever It -xl :S O. Thw, the right side of (1.37) is within E of f(x) whenever h E (0,8] Thus the detired limit exists as h \, O. A similar argument treats h / O. The next result is the rest of the Fundamental Theorem of Calculus. Theorem 1.7. If G is differentiable and G (x) ~s contmuous on [a, b], then (1.38) .ib G (t) dt = G(b) - G(a). Proof. Con -a). whNe / l is the right side of (1.41), we can a::,::,ume without lo::,s of generality that f(a) = f(3) Then we claim that f (~) = 0 for ::,ome c: E (a,.3) Indeed, since [a, f:I] is compact, f mu::,t as::,ume a maximum and a minimum on [a, 3] If f(a) = f(8), one of these mu::,t be assumed at an interior point. C:. at which f c