An uncountable family of regular Borel measures on

arXiv:math/0703117v1[math.FA]5Mar20071AnUncountableFamilyofRegularBorelMeasuresOnCertainPathSpacesofLipschitzFunctions.01/19/07R.L.BakerUniversityofIowaIowaCity,Iowa522422AnUncountableFamilyofRegularBorelmeasuresR.L.BakerUniversityofIowaIowaCity,Iowa522423ABSTRACT.Letc0beafixedconstant.Let0≤rsbeanarbitrarypairofrealnumbers.Leta,bbeanypairofrealnumberssuchthat|b−a|≤c(s−r).DefineCsrtobethesetofcontinuousreal-valuedfunctionson[r,s],anddefineCrtobethesetofcontinuousreal-valuedfunctionson[r,+∞).Finally,considerthefollowingsetsofLipschitzfunctions:Λsr={x∈Csr||x(v)−x(u)|≤c|v−u|,forallu,v∈[r,s]},(1)Λr={x∈Cr||x(v)−x(u)|≤c|v−u|,forallu,v∈[r,+∞)},(2)Λs,br,a={x∈Λsr|x(r)=a,x(s)=b},(3)Λsr,a={x∈Λsr|x(r)=a},(4)Λs,br={x∈Λsr|x(s)=b},(5)Λr,a={x∈Λr|x(r)=a}.(6)WepresentageneralmethodofconstructinganuncountablefamilyofregularBorelmea-suresoneachofthesets(1),(2),andanuncountablefamilyofregularBorelprobabilitymeasuresoneachofthesets(3)-(6).Usingthismethod,wegiveadefinitionofLebesguemeasureonthesets(1)and(2),andadefinitionoftheuniformprobabilitymeasureoneachofthesets(3)-(6).Keywords:infinitedimensionalLebesguemeasure,Lipschitzfunctions,Radonmeasures,uniformprobabilityprobabilitymeasure.MathematicalReviewssubjectclassification:26A99,28C05,28C15,28C20,60G05,81S40.41.INTRODUCTIONLetc0beafixedconstant.LetRbethesetofrealnumbers.ForeachintervalI⊆R,letΛ(I)bethesetoffunctionsx:I→RsuchthatxsatisfiestheLipschitzcondition|x(t)−x(s)|≤c|t−s|,forall,s,t∈I.Letr,sbeanypairofrealnumberssuchthat0≤rs.DefineΛsr=Λ([r,s]),Λr=Λ([r,+∞)).Finally,leta,bbeanypairofrealnumberssuchthat|b−a|≤c(s−r).ThendefineΛs,br,a={x∈Λsr|x(r)=a,x(s)=b};(1)Λsr,a={x∈Λsr|x(r)=a};Λs,br={x∈Λsr|x(s)=b};Λr,a={x∈Λr|x(r)=a}.Themainresultofthepresentpaper(Sections2and3)isageneralmethodofconstructinganuncountablefamilyofregularBorelmeasuresλs,br,a,λsr,a,λs,br,λr,a,(2)λsr,λr,(3)respectively,oneachofthefollowingsets:Λs,br,a,Λsr,a,Λs,br,Λr,a;(4)Λsr,Λr.(5)EachBorelmeasurein(2)isaprobabilitymeasure,constructedasthecontinuousimageofLebesguemeasureunderacertainfamilyofcontinuoussurjectivemappingsϕs,br,a:Ωsr→Λs,br,a,ϕsr,a:∧Ωsr→Λsr,a,ϕs,br:∨Ωsr→Λs,br,∧Ωr→Λr,a.(6)EachofthespacesΩsr,∧Ωsr,∨Ωsr,∧Ωr(7)isendowedwithLebesguemeasure,andhastheform[0,1]A,whereAissomeindexingset.Likewise,eachBorelmeasurein(3)isconstructedasthecontinuousimageofLebesguemeasureunderacertainfamilyofcontinuoussurjectivemappingsϕsr:∼Ωsr→Λsr,ϕr:∼Ωr→Λr,(8)whereeachofthespaces∼Ωsr,∼Ωr(9)hastheform[0,1]BforsomeindexingsetB,andisendowedwithLebesguemeasure.5InSection4,certainmembersλs,br,a,λsr,a,λs,br,λr,aoftheuncountablefamily(2)aresingledoutanddefinedtobetheuniformprobabilitymeasureonthespacesΛs,br,a,Λsr,a,Λs,br,Λr,a,(10)andcertainmembersλsr,λroftheuncountablefamily(3)aresingledoutanddefinedtobeLebesguemeasureonthespacesΛsr,Λr.(11)2.CONSTRUCTIONOFTHEFUNCTIONSϕInthissectionweconstructthefamiliesofcontinuoussurjectivemappingsmentionedin(6)and(1)oftheintroduction.Definition2.1.Let0≤rsbegiven.Let(r,a),(s,b)betwogivenpointsintheplane.DefineFr,a={(t,x)|r≤tanda−c(t−r)≤x≤a+c(t−r)},Bs,b={(t,x)|t≤sandb−c(s−t)≤x≤b+c(s−t)},Ps,br,a=Fr,a∩Bs,b.Proposition2.1.Forarbitrarypairs(r,a),(s,b),with0≤rs,wehavePs,br,a6=∅ifandonlyif|b−a|≤c(s−r).IfPs,br,a6=∅,thenPs,br,aiseitherthelinesegmentconnecting(r,a)to(s,b),orPs,br,aisanondegenerateparallelogramcontainingthislinesegment.Proof.Theproofofthispropositionisroutine.Definition2.2.AssumethatPs,br,a6=∅.Letu=12(r+s).DefineIs,br,atobetheprojectionofthefollowingsetontothex-axis.{(u,x)|−∞x+∞}∩Ps,br,a.BecausePs,br,aisaparallelogramcontainingthelinesegmentjoining(r,a)to(s,b),andbecauserus,weseethatIs,br,aiseitherapointoranondegenerateclosedinterval.6Proposition2.2.TheintervalsIs,br,aaregivenbyIs,br,a=[b−12c(s−r),a+12c(s−r)],ifa≤b;[a−12c(s−r),b+12c(s−r)],ifb≤a.Proof.Theproofofthispropositionisclear.Definition2.3.Weshallassumethatforeachpair(r,a),(s,b)ofpointsintheplanesuchthat0≤rsandPs,br,a6=∅,wearegivenacontinuousfunctionλs,br,a:[0,1]→Is,br,amapping[0,1]ontoIs,br,a.Weshallalsoassumethatthemapping(a,b,r,s,ξ)7→λs,br,a(ξ)iscontinuousonthesetDλ,whereDλ={(a,b,r,s,ξ)∈R5|0≤rs,|b−a|≤c(s−r),andξ∈[0,1]}.Proposition2.3.AssumethatPs,br,a6=∅.Letu=12(r+s).Ifd=λs,br,a(ξ),whereξ∈[0,1]isarbitrary,then∅6=Pu,dr,a,Ps,bu,d⊆Ps,br,a.Proof.Wewillonlyprovethat∅6=Pu,dr,a⊆Ps,br,aandPu,dr,a⊆Ps,br,a.Therestoftheproofissimilar.Also,weonlygivetheproofforthecasewherea≤b,theproofforthecaseb≤aissimilar.WethehaveIs,br,a=[b−12c(s−r),a+12c(s−r)].Becaused=λs,br,a(ξ),itfollowsfromDefinition2.3thatd∈Is,br,a,thereforeb−12c(s−r)≤d≤a+12c(s−r).(1)ByProposition2.1,toprovethatPu,dr,a6=∅,itsufficestoprovethat|d−a|≤c(u−r).(2)Notethatu−r=12(s−r).Hence,from(1),d≤a+12c(s−r)=a+c(u−r).Italsofollowsfrom(1)thatd≥b−12c(s−r)=b−c(u−r)≥a−c(u−r).Weconcludethata−c(u−r)≤d≤a+c(u−r).7Therefore,|d−a|≤c(u−r),hence(2)holds.ToprovethatPu,dr,a⊆Ps,br,a,let(t,x)∈Pu,dr,abearbitrary.Thenbydefinition,r≤t≤sanda−c(t−r)≤x≤a+c(t−r),d−c(u−t)≤x≤d+c(u−t).(3)Wehaveu−r−s−u,andby(2),d≤a+c(u−r),consequently,(3)impliesthatx≤d+c(u−t)≤[a+c(u−r)]+c(u−t)=a+c(s−u)+c(u−t)≤b+c(s−u)+c(u−r)=b+c(s−r).Thus,x≤b+c(s−r).