AN APPLICATION OF NONSTANDARD ANALYSIS TO
CHARACTERS OF GROUPS OF CONTINUOUS FUNCTIONS
著者
KAWAI Toru
journal or
publication title
鹿児島大学理学部紀要. 数学・物理学・化学
volume
12
page range
43-45
別言語のタイトル
連続函数の群の指標への超準解析の応用
URL
http://hdl.handle.net/10232/6374
AN APPLICATION OF NONSTANDARD ANALYSIS TO
CHARACTERS OF GROUPS OF CONTINUOUS FUNCTIONS
著者
KAWAI Toru
journal or
publication title
鹿児島大学理学部紀要. 数学・物理学・化学
volume
12
page range
43-45
別言語のタイトル
連続函数の群の指標への超準解析の応用
URL
http://hdl.handle.net/10232/00003971
Rep. Fac. Sci. Kagoshima Univ., (Math., Phys. & Chem.), No. 12, p. 43-45, 1979
AN APPLICATION OF NONSTANDARD ANALYSIS TO CHARACTERS
OF GROUPS OF CONTINUOUS FUNCTIONS
By Toru Kawai* (Received September 29, 1979) LetKbeacompacttotallydisconnectedspace,andletS(K)bethemultiplicative groupofallcontinuouscomplex-valuedfunctions/onKsuchthat￨/I-1.Thegroup S(K)isatopologicalgroupunderthemetric d(f,g)-sup¥f(k)-g(k) kesK(f>9∈S(K)). In[3]and[4],VaropouloshasdiscussedcontinuouscharactersofS(K)(seealso [1]).Projectivelimittechniqueshavebeenusedin[4].Applyingnonstandard methods,wegiveasimpleandnaturalproofofVaropodos's也eorem. ● WeuseanonstandardsettheoryNSTwith,anaxiomsystemin[2].Insteadof比e axiomschemaofsaturation([A.5]in[2.]),wemayadopttheaxiomschemaofenlarging ([A.5E]in[2]),whichisweaker.Whicheverwechoose,everystandardin丘nite sethasnonstandardelements.LightfaceLatinlettersdenotestandardsets,andGreek lettersdenoteinternalsets. Theorem(Varopoulos).LetFbeacontinuouscharacterofS(K).Thenthereexist a動itenumberofpointskv ,kj∈andintegersp(l),-,p(J)∈Zsuchthat F(g)-n[g(研tf>forallg∈S(K). Proof.LetgbethecollectionofallfinitepartitionsofKintonon-emptyopen closedsubsets.ForDl9D2∈&,wewrite蝣 Dl≦jD2ifforeveryA∈D望thereisB∈Dl suchthatAcB.Sincetherelation"^"isconcurrent,thereisaninternalpartition A∈&suchthatD≦AforanystandardD∈^.ForeachD∈let KD:[1,¥D¥]-D beabisection,andlet xD:[l,¥D¥l-K beachoicefunctionsuchthatxD(m)∈KD(m)(1≦m≦lDI),wherelD¥isthecardinal numberofD,and[1,￨D￨]istheintervalinZ.SinceKistotallydisconnected,the propertyofAshowsthatthestandardpartofKA(fjL)isasingletonsetforanya牀 [1,刷].ForeachD∈wedefine
44 T. Kawai A-{g∈S(K):gisconstantoneachpartitionsetofD) andSD-*A,where*Aisthestandardsethavingthesamestandardelementsasthe ■ externalsetA.EorD∈&,therestrictionFlopisacontinuouscharacterofSD. SinceSDistopologicallyisomorphictothe¥Dトdimensionaltorusgroup,thereisa ● mapping *:[1,IDt]-Z suchthat ¥D¥ W)-n[f(xD(m))]'(叫forall/∈SD･ Bythetransferprinciple,thereisaninternalmapping ●● 入:[1,IAf]-Z suohthat
FU)-餅Mp))]*<*> for all <f>∈SA･
(1)41 MI supposethatQ-II人中0.DefineafunctionifjeS^by ip(x4(i))-exp(isgn入(p)IQ)(p牀[l,刷])･ Then(1)showsthatF(ip)-ei.SinceFiscontinuous,舶1)isnotinfinitesimal.This impliesthatthepositiveintegeriiisfiniteandishencestandard.Thus入(a)isa standardintegerforeach/lc∈[1,刷],and tFL∈[1, Ml]:A(p)串0)
is a finite set of internal positive integers. Therefore (1) can be rewritten as
F(<f)- n U(Kj)]HJ) for all <f>*s*.
(2)where / is a standard natural number, p(j) are standard integers and ka are internal elements ofK. For eachj, let {kj} be the singleton set which is the standard part of the partition set of A containing k;-. Then we have kj剛(1≦j≦J). Ifォ∈S{K), then there is a ≠∈Sj such that d(g, <f>)-0. This implies that
･f>(Kj) **9¥Ki) -gih)蝣
It follows from continuity of F that F(g)-F(J>). Taking the standard part in (2), we have
∫
2%)- n [<7(W<サ.
ノ=1
Nonstandard Analysis and Groups of Continuous Functions
● 45
References
[lj Bernard, A. et Varopoulos, N. Th., Groupes de fonctions continues sur un Compact.
Applications a Vetude des ensembles de Kronecker, Studia M二ath., 35 (1970), 199-205.
[2] Kawai, T., An axio㈹ system for nonstandard set theory, Rep. Fac. Sci. Kagoshima Univ.
(Math., Phys. & Chem.), 12 (1979), 37-42.
[3] Varopoulos, N. Th., Sur les ensembles de Kronecker, C.R. Acad. Sci. Paris, Ser. A, 268
(1969), 954-957.
[4] Varopoulos, N. Th., Groups of continuous functions in harmonic analysis, Acta Math., 125 (1970), 109-154.
