Some Applications Of The Bounded Convergence Theorem For An .

Some Applications of the Bounded Convergence Theorem for an Introductory Course inAnalysisAuthor(s): Jonathan W. LewinSource: The American Mathematical Monthly, Vol. 94, No. 10 (Dec., 1987), pp. 988-993Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2322609 .Accessed: 09/01/2015 11:53Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at ms.jsp.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.http://www.jstor.orgThis content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

988JONATHANW. eApplicationsAnalysisinCoursean IntroductoryJONATHANW. LEWINKennesawCollege,Marietta,GA 30061is thespecialcase of theLebesguetheoremThe Arzelaboundedconvergenceareassumedtobe nceintegrable.THE BOUNDED CONVERGENCETHEOREM. Suppose (fn) is a sequence offunctionson an interval[a, b], supposethatthesequence(fn)whichare Riemannintegrablef, and supposethatthereexistsa numberK suchthatconvergespointwisetoa functiondxEandx c [a, b]. ThenthesequenceofintegralsnZ allKforIf"(x)I fabfn(x)on [a, b], wef is also Riemannintegrableconverges,and in theeventthatthefunctionhavefnb(x)dx -Xf (x)dx.as quitehard,beenperceivedhas remcoursein analysis,whichliebeyonda firston conceptsor dependentat thislevelhavethereforeomittedin suchcourses,and attheHowever,a recentbeensomewhatneglected.theoremcanbe provedquiteeasilyin a firstcourse,and it isboundedconvergencebe. In thispaperwe foretheoremmaybe used to obtainsimpleproofsofhow iationsome quitesharpformsof thetheoremsof theseWe shallobtainversionsofrepeatedintegrals.signand inversionintegralwhichare distinctlysharperthantheresultsusuallyfoundin an undertheoremsgraduatetext.undertheintegralsign.In a typicalfirstcoursein analysis,theDifferentiationfuncundertheon differentiationintegralsignare givenforcontinuoustheorems10 (see,forexample,it is easy to dropthetheorem,7.40). However,usingtheboundedconvergenceof the typeone mightand obtainsharpertheoremsof continuity,requirementA theoremof theexpectto see at a remsharpertypemaybe foundin [1,Theorem10.39],threechapters7.40,in thechapteron Lebesgueintegration.STHEOREM ON DIFFERENTIATING UNDER THE INTEGRAL SIGN. Suppose f: [a, b] X-*R, whereS c R, and thatforeverypointy c S, theRiemannintegral(Y) Jf(x, y) dxexists. Suppose yo is botha point of S and a limitpoint of S, and thatfor everyThis content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

1987]989THE TEACHING OF MATHEMATICSx E [a, b], thepartialderivativeD2f(x, YO) lim [[f(X Y)Y YoY-f(x, YO)]YoJexists,and supposethattheRiemannintegralfaD2f (x, yo) dx exists.Supposefinallythatthereexistsa numberK such thatforall x E [a, b] and y E S\ {Yo ), we have[f(X, Y) -X, Yo)K.ThenP (yo) JbJf(XIyo) dxProof.We theorem.Givenanysequence(yn)in S \ { yo), convergingto yo,we have- Pyf(X,YJ -f(x, b[f(x,yn)f Yo)bf D2f(x,yo)dx (Yn) OAdxas n oo.Yn YJaYn YOA posef: [a, b] X SRiemannintegral-9R, whereS is an interval,and thatforeverypointy E S, the (y) fb(xI y) dxexists. Suppose thatfor everyx E [a, b] and for everyy E S, thepartial derivativeD2f(x, y) exists, and that the RiemannintegralfaD2f(x, y) dx exists. Supposefinallythatthereexistsa numberK, suchthatforall x E [a, b], and y E S, we haveID2f(x, y)I K. ThenforeveryyE S, wehave '(y) D2f (xy) dx.The usefulanaloguesof thistheoremforimproperRiemannintegralscan bededucedalmostas simply,usingan obvious"dominatedconvergence"analogueofthe boundedconvergencetheoremwhichwouldapplyto improperRiemannintegrals.As an exampleof the sortof resultthatcan be obtained,we cite thefollowing:THEOREM ON DIFFERENTIATING AN IMPROPER INTEGRAL UNDER THE INTEGRALSIGN. Suppose - oo a b oo, S is an interval,and thatf: [a, b) X S -R.Suppose thatforeverypointx E [a, b), thefunctionf (x,) is differentiableon S, andthatforeverypointy E S, thefunctionsRiemannft( , y) and D2f(, y) are improperintegrableon [a, b), and supposefinallythat thereexists an improperRiemannintegrablefunctiong on [a, b) such thatfor all x E [a, b) and y E S, we havey)IID2Af(x, Ag(x)This content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

990[DecemberJONATHANW. LEWINThenifwe define(Y) bf(X, y)dxfor all y E S, we have(y y) dxbD2f(x,at everypointy E S.Inversionof repeatedintegrals.The sharpestknownresulton inversionofiteratedRiemannintegralsis the elegantresultthatwas provedin 1913 by G.Fichtenholz.We shallstatethreeversionsof Fichtenholz'stheorem.The firstoftheseis the easiestto prove,thesecondis ,and thethirdformis theultimatetheoremon theinversionofiteratedintegralsfora boundedfunctiondefinedon a rectangle.In thisthirdformof thetheorem,we see thatthetheoremremainstrueevenifsomeof theintegralsare onlyassumedto be Lebesgueintegrals.FICHTENHOLZ'STHEOREM ON INVERSION OF ITERATED INTEGRALS FIRST FoRM.Supposef is a boundedfunctionon therectangle[a, b] X [c, d ]. Thentheidentity(bx,fy) dydx fd f (x, y) dxdywillholdifbothsidesexistas repeatedRiemannintegrals.SECOND FORM. Supposef is a boundedfunctionon therectangle[a, b] x [c, d].Suppose thatforeverypointx E [a, b], thefunctionf (x,.) is Riemannintegrableon[c, d ], and thatforeverypointy E [c, d ], thefunctionf(., y) is Riemannintegrableon [a, b]. Then(a) Thefunction4: [a, b] -) R definedby4)(x)is Riemannintegrableon [a, b],The4:d(b)function [c, ] -) R definedby (y)is Riemannintegrableon [c, d], dx(c) fab?4((x)fJ4(y) dy, in otherwords,bfdf (x y) dydx fJdf(x, y) dyforall x (dx, fe [a, b],fabf(x,y) dxforally e [c, d],y) dxdy.THIRD FoRM. Supposef is a boundedfunctionon the rectangle[a, b] X [c, d].Suppose thatfor everypointx E [a, b], thefunctionf (x, ) is Riemannintegrableon[c, d ], and thatforeverypointy E [c, d ], thefunctionf(, y) is Lebesguemeasurableon [a, b]. ThenR definedby4 (x) fdf(x, y) dyforall x e [a, b],(a) Thefunction4: [a, b]is Lebesguemeasurableon [a, b],(b) Thefunction p:[c, d] -) R definedby (y) fabf(x, y) dx forall y e [c, d],is Riemannintegrableon [c, d],This content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

1987]THE TEACHING991OF MATHEMATICS(c) fab (x) dx ftd'(y) dy, in otherwords,fbdf(xI y) dydx Jdfbf(x, y) dxdy.Proofofthefirstform.For eachnaturaln, denoteas ?ntheregularn-partitionof [c, d]. For i 1,. . ., n, theith pointof gn is,of course,c i(d - c)/n, but forwe shalldenotethisas yni.For eachnaturaln and x e [a,b],definesimplicity,n(An(x)f(x,Yni)(YnifYni-1)i l1Since the functionf(x, ) is Riemannintegrableforeveryx E [a, b] and since0,it followsfromDarboux'stheoremthaton(x) -4 (x) foreach x E [a, b].II nSincewe havealso assumedthat4 is Riemannintegrableon [a, b],it dx fb()fbff(x,y)dydxdasnoo.Butforeach n, we havejb4n(x)dx , f(x,i- 1 [Yni)(Ynif(x?Yni)-Yni-1) dxdx] (Yni -Yni-i1)nEi 1 (Yni)(YniYni-1)and since 4 is Riemannintegrableon [c,d], the latterexpressionapproachesf/4l(y) dy as n -4 so.Thisshowsthatfab (x) dx fcd"(y)dy whichis whatwe had to ofthesecondform.betweenand thefirstis d 4 is nowpart of the conclusion.Whatwe have to showtherefore,is that4) and 4 areRiemannon [a, b] and[c,d], respectively.automaticallyintegrableAs above,let9/nbe theregularn-partitionof[c,d] foreachnaturaln, and denotetheithpointofintegrableon [c, d], we shallshowthattheregn as yni.To showthat4 is Riemannis a numberL suchthatforeverypossiblechoiceof numberstni in theintervalsyni-1yniwe havenEi 1-)(tni)(Yni-Yni Las n -oo.Let us look forthemomentat one possiblechoiceof thenunberstni,.For eachThis content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

992[DecemberJONATHANW. LEWINnaturaln and x E [a, b],definenOn(X)i lf(x tni)(Yni Yni-1)fSince the functionforeveryx E [a, b] and sincef(x, ) is 0,IInIIj on(x) - 4)(x) foreachx E [a, b].It, hatthesequenceofintegralsfn!'4)(x) dx converges.The limitof thissequenceofintegralsis obviouslyindependentof the choiceof numberst forif t*1is anotherchoice,and thefunctionsn*aredefinedanalogouslyby*(X) ni 1f(x, t,i)(yni-yny,1) for x e [a, b],thenwe also have *(x) - O(x) forall x E [a, b] and theboundedconvergencetheoremdx - 0. Now foreachn, we haveimpliesthatJfa[On(x) - O)n*(x)]aIn(X) dx Ei ndsto a limitas required.Thisshowsthat44is Riemannintegrableon [c,d]. Theproofthat is Riemannon [a, b] isintegrablesimilar.Proofofthethirdform.As in theproofofthesecondform,we needto showthat4 is Riemannintegrableon [c, d]. Theproofwe use nowis similarto theone usedbeforeexceptthatthistime,we have to makeuse of the Lebesguedominatedconvergencetheorem.As before,denoteas ?P,7theregularn-partitionof[c,d] andthe ith pointof 9,, as y,,i.We shallprovethe theoremby showingthatcp ismeasurableon [a, b], and thatforeverypossiblechoiceof numberstniin theintervals[y,,i-,,y,,j]forn - 1,. and i 1,., n, we have4(x) dx.(tli)(Y"i-Yni-1) aSuppose then,thatthenumberst,nihavebeen chosen.For each naturaln andx E [a, b], definei-1)1oil (X xi lf (X, tni ) (Yni-Yni-1)and noticethateachfunctionof Lebesguemeasur4,, beinga linearcombinationable functions,is Lebesguemeasurableon [a,b]. As above,it followsfromtheof thefunctionsRiemannintegrabilityf(x, *) that jx(x)-* O(x) foreach x E[a, b]. Therefore4i is Lebesguemeasurableon [a, b] and it nceb (x) dxbj (x) dx - This content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

1987]THE TEACHINGOF MATHEMATICS993and theresultfollowsas beforefromtheidentityjA f(x)adxi 1tni)(YniYni-1)An interesting(and possiblysurprising)featureof Fichtenholz'stheoremis thefactthatit makesno requirementofintegrabilityoff jointlyin thetwovariablesxand y. The theoremin erentand fromthetheoremson pages111-114of Buck[2] and thosein Section7.25ofif theContinuumApostol[1]. As is wellknown,is assumed,thentheHypothesisforanalogueof Fichtenholz'stheorem Lebesgueintegrals-is not even true;seeRudin[5,page grabilityofthefunctionwithrespectto at leastoneofitsvariablesis reallyneeded.Somefurthercounterexamplesmaybe foundin Luxemburg[4],whichalso containsasignificantof Fichtenholz'sgeneralizationtheoremto some abstracttheoriesofButit shouldbe mentionedintegration.thatone of theexamplescitedby Luxemof Propositionburgis incorrect,possiblya resultof a misreadingC49 in Sierpiniskicitestheincorrecta[6]. Luxemburgascountertoexampleexample theabovethirdformof eading,1. Tom leyMassachusetts,1974.NewYork,1965.2. R. any,theorem,American3. JonathanW. Lewin,A trulyelementaryapproachto theboundedconvergenceMathematicalMonthly,93 (1986) 395-397.TheAbstractRiemannanda TheoremofG. Fichtenholzon Equality4. W. A. J.Luxemburg,IntegralIA and IB, Proc. Ned. Akad. Wetensch.Ser. A 64 (1961)of RepeatedRiemannIntegrals,516-545 Indag. Math.,23 (1961).McGraw-HillBookCompany,NewYork,1974.5. WalterRudin,Real andComplexAnalysis,du Continu,Warsaw(1934),second.ed.,NewYork,1956.6. Sierpifnski,W.,HypotheseThis content downloaded from 69.91.134.221 on Fri, 9 Jan 2015 11:53:52 AMAll use subject to JSTOR Terms and Conditions

Some Applications of the Bounded Convergence Theorem for an Introductory Course in Analysis JONATHAN W. LEWIN Kennesaw College, Marietta, GA 30061 The Arzela bounded convergence theorem is the special case of the Lebesgue dominated convergence theorem in which the functions are assumed to be Riemann integrable. THE BOUNDED CONVERGENCE THEOREM.