An Introduction to Nonstandard Real Analysis (Pure and Applied Mathematics (Academic Press), Volume 118)

247 Pages·1985·9.44 MB·English

by  Albert E. Hurd

AnI ntrodutcot ion
NonstanRdeaaArlnd a lysis
Thiiss a volumie n
PURE AND APPLIMEADT HEMATICS
A SerieMso noofg rapahnsdT extbooks
EditoSrAs:M UEELl LENBAENRDHG Y MABNA SS
Al i�t otfi trilentec hsesi nestr aipepesa artst heen odf t hviosl ume.
AnI ntrodutcot ion
Nontndard
s a
RelaA nlaysis
ALBERT E. HURD
DepartmoefnM ta thematics
UniverosfiV tiyc toria
VictorBirai.t iCsohl umbia
Canada
PETER A. LOEB
DepartmoefnM ta thematics
UnviersoiftyI llinois
Urbana1,/ linois
1985
ACADEMIC PRESS, INC.
(HarcoBurratc Jeo vanovPiuchb,l ishers)
OrlandSoa nD iegoN ew York London
TorontMoo ntreaSly dneyT okyo
COPYRIGHTIIC>985B YA CADEMIPCR ESSI,N C.
ALLR IGHTRSE SERVED.
NOP ARTO FT HIPSU BLICATIONB EMR AEYP RODUCOERD
TRANSMITTEIDN ANFYO RMO RB YA NYM EANSE.L ECTRONIC
ORM ECHANICAILN.C LUDIPNHGO TOCOPYR.E CORDINOGR,A NY
INFORMATIOSNTO RAGEA NDR ETRIEVSAYLS TEWMI.T HOUT
PERMISSIIONWN R ITINFGR OMT HEP UBLISRH.E
ACADEMPIRCE SISN,C .
Orlando, Florida 32887
UnitKeidn gdom Editiobny published
ACADEMPIRCE SISN C(.L ONDOLNT)D .
24/O2v8a Rloa dL,o ndNoWn I 7DX
LibroaryfC ongress CianPt ualbolgiiDncagat tai on
Maienn tryun dert itle:
An introducnt tioo nonstandarreda la nalysis.
Includbeisb liographircefaelr enceasn di ndex.
1.a thMematicala nalysisN,o nstandar1d,.H urd,A . E.
(AlberEtm ers)o,n DATE • II.o ebL,P . A.
QA299.82.115988 5 515 84-24563
ISB0N- 12-36244(0a-11k .p aper)
PRINTIEND TUHNEI TSETDA TI::.S OF AME:RICA
85 86 8887 987654321
Dedicattoe dmt ehmeo royf
ABRAHARMO BINSON
ThiPsa geI ntentioLneaflBtll ya nk
Contents
Prefa.c e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . ... . . . .. . . . .. . . . . . . . . . . 1x
ChaptIe r
lnfiniteasnidmT ahleCs a lculus
I.I TheH yperreaNlu mberS ysteamsa nU ltrapowe..r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 •-TransfoofrmR se lati.o..n..s... ....................................................... 8
1.3 Simple LangfuoarRg eelsa tional S.y s.t .e m.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
1.4 InterpretaotfiS oinm plSee ntenc...e.s.. .............................................. 15
1.5 TheT ransfPerri ncifpolrSe i mplSee ntenc..e..s.. .................................. 19
1.6 InfiniNtuem bersl,n linitesiamnadtl hse,S tandaPradrtM ap. ....................2 4
I7. TheH yperinteg. .e r.s . . . . . . . . . . . . . . . .2 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 SequencaensdS erie..s.. ............................................................... 32
1.9 Topologoyn t heR eal.s.. ............................................................... 39
I.1 0 Limiatnsd C ontinu.i...t.y... ........................................................... 44
II.I Differentia. . t. i. .o ...n .. . ... . . . . . . . . . ... .. . . . . . ... ... . . .. . .. .. ... . . . ... ... .. ... .. . .. ... . . 51
1.12R iemanInn tegiroan.t ...................................................................5 6
I1.3 SequencoefFs u ncti.o..n..s. ........................................................... 60
1.14 Two ApplicattiooD nisf fereEqntuiaatli o.n..s.. .................................... 63
1.15P rooofft heTr ansfPreirn cip..l..e... ................................................. 67
ChaptIeIr
Nonstandard AonnaS luypseirss trreusc tu
111. Superstcrutur.e.s.. ....................................................................... 71
11.2L anguagaensdI nterpretatfioorSn u perstruct..u..re..s... .......................... 74
113. Monomorphisbemtsw eenS uperstruurcetsT:h eT ransfPerirn cip.l..e.. ......... 78
11.4T heU ltrapoCwoenrs tructfioorSn u perstructu..r..e..s.. .......................... 83
11.5H yperliniSteet sE,n largemeanntdCs o,n curreRnetl ati.o..n.s.. .................8 8
116. Intelra nnadE xternaEln titiCeomsp;r ehensiv.e..n..e...s..s.. ..................... 94
11.7T heP ermanence Pr.i..n...c.i...p...l.e.. ............................................ 100
11.8K -SaturSautpeerds tructu. .re.s . .. . . . . . . .. . .. .. . . . .. . . ... .. .. .. ... .. . . . . . .. .. . .. .. . . . . 104
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viii Contents
ChaptIlel r
NonstandTahredo royfT opologSipcaacle s
III.I BasiDce finitainodRn ess ul.t s. . . . . . . . . . . . . . 1.1 .0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111.C2o mpactsnse . . . . . . . . . . . . . . . . . . . . 1.2 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II3I .M etircS pace.s . . . . . . . . . . . . . . . . . . .1 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II4I .NormeVde ctoSrp acsea ndB anachS pace.s . . . . . . . . . . .1 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . .
111.I5n ner-PrtoS dpuacceasn dH ibe rt Space.s ... ..... ... . .. ..... . ..... . ...... .... 1.4 .5 .. ..
111.N6o nstandHaurldlo sf M etric Sp.a c.e s. . . . . . . . . . .1 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II.7 I Compactifica. .t i.o n.s . . . . . . . . . . . . . . . 1.5 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111.F8u nctiSopna ce.s.. ................................................................... . 160
ChaptIeVr
NonstandaIrndt egraTthieoonr y
IV.I StandardizoaftI inotnesrni anlt egraSttirounc tu..r ..e. s . .... . . .. ... . . . .1..6 5.. .. .. . .
IV. 2 MeasurTeh eorfyo rC ompleItnet egraSttiruocnt ur. .e s. . . . . . . 1.7 5. . . . . . . . . . . . . . . . . .
IV. 3I ntgreatioonnR "t;h eR iesRze presentTahteioorne ..m . . . . ..... .. 1..8 9.. . .. . . . . . . . .
IV. 4 Basic Convergence. .T h.e o.r e.m s. . .. . .. .. . . 1. 95. . . . . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . . ..
IV.5 TheF ubini The.o.r.e.m. .......................................................2.00. ....
IV.6A pplicattioSo tnso chasPtrioccs eses.. .............................................2 05
Appendix
Ultrafi..l...t..e..r..s... ............................................................... 219
References 222
List of .S...y....m..b...o...l...s... .... ..................................................... 225
Inde..x.. .................................................... ...... ............................... 227
Preface
Then otioofna ni nfinitehsaisam papela reodf fa ndo ni nm athematsiicnsct eh e
timoef A rchimedIensh .i sf ormulaotifto hnec alcuilnut sh e1 670tsh,eG erman
mathematiWciilahne lGmo ttfrLieeidb ntierza teidn finitesiamsia dlesa nlu m­
bersr,a thleirk iem aginanruym berwsh,i cwhe res mallienar b solute value than
anyo rdinarreyan lu mberb utw hicnhe vertheolbeesysea dl olf t heu sual loafw s
arithmeLteiicb.n riezg ardiendf initesaisam uaslesf fuilc tiwohni cfha cilitated
mathematciocmaplu tatainodin nv entiAoln.t houigtgh a inerda piadc ceptaonnc e
thec ontinoefnE tu ropeL,e ibnimze'tsh owda sn otw ithoiuttds e tractIonr s.
commentionngt hef oundatioofcn asl culasud se velopebdo thb yL eibnainzd
NewtonB,i shoGpe orgBee rkelweryo te",A ndw hata ret hessea mee vanesncte
incremenTthesy?a ren eithfeirn iqtuea ntitnioerqs u,a ntiitnifiensi tsemlayl nlo,r
yetn othinMga.y w e notc altlh emt heg hosotfsd eparteqdu antitiTehseq ?u"e s­
tiowna s,H ow cant herbee ap ositniuvmeb ewrh icihs s malltehra ann yr eal
numbewri thobueti nzge roD?e spittheiu sn answerqeude stitohnei, n finitesimal
calcuwlaussd evelopbeyEd u learn do thedrusr nigt hee ightntehea ndn ineteenth
centuriinetason i mpressbiovdeyo fw orkI.tw asn otu nttihlle a tnei netntehec en­
turtyh aatna dequate deoffil niimtriietop nl actehdce a lcuolfui sn finitesainmda ls
providaer di gorofouusn datfiooarnn alysis. Following thtihseu sdee velopment,
ofi nfinitesgirmaadlusa flaldye dp,e rsistoinnlgya sa n intuitaiivdte o c on­
ceptualization.
Therteh e matsttoeorud n ti1l9 60w henA brahaRmo binsgoanv ea rigorous
foundatfioortn h eu seo fi nfinitesinia mnaallsy sMiosr.es pecificaRlolbyi,n son
showetdh atth es eotf r eanlu mbercsa nb er egardaesad s ubsoefta largseerot f
"number(sc"a llheydp errenaulm berwsh)i ccho ntaiinnsf initesainmdaa lloss,
witahp propridaetfeilnyae rtdi thmeotpiecr atisoantsi,s afiloelfs t hea rithmetic
ruleosb eyebdy t heo rdinarreyan lu mberEsv.e nm oreh,e d emonstrtahtaettdh e
relatisotnraulc tovuerret her eal(ss etrse,l atieotncsc.,a) n b ee xtendteoad sim­
ilasrt ructouvreetr h eh yperreiansl usc ahw ayt haatl slt atemetnruteis n t her eal
structruermea itnr uew,i tah suitaibnltee rpretaitnti hoenhy ,pe rreals tructure.
Thilsa ttperro pertkyn,o wnas t hter anspfreirn ciipstl hepe,i votraels uolfRt o bin­
son'dsi scovery.
ix