collection - w3.impa.brw3.impa.br/~monets/notes/cap7.pdf · cau niekaub lcb ndecan b c v bin socau...
TRANSCRIPT
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Completedleasures
Defittmeasurablespace is a couple Diewhere Disasetanotoelisao algebraofsubsetsof A A measure space is a tripleCAGmwhere Afd is a measurablespaceandµ is ameasuredefined in TeDef A measure space is saidtobecompleteifAERMAKO BE A Bede
Obs In that case µlBI 0Def LetCaffee µ be a measure space Fakeclassofnullsetsassociated to CATe a is thecollectionN NEA FAEfee NEAandul Ako
Prop Nisa o ring i.ea NNEN NIN EN
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b Ni e N ie IN µNi E N
Demi a LetNs Nz e NZhenthereexistsAcSeesuchthattheAand µ A o FoherforeNi NE AandNi Ni eNb LetNieN ie INZhenthereaistAieTe ie INsuchthatNi eAiandeefAit 0 forany ie IN 8hereforeithNi E nAi andMiunAi 0
soiN i E N
DefZhecompletionof 8 underµ is theclassofsetsoftheform
TI AuN AcSee Ne NPropiTet is a o algebraDemi a It is clearthat loDne atb EakeA ete NE N Cohenthereaists Be AsuckthatNEBand µCBI O Cohen
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CAu NI EKAuB lCB NDECA n Bc v BINso CAu NYETIc Fake Ai EAl Ni e N i e IN ZhenthtAiuNi l fivef i uiuenNilandtherefore afAiuNiI e TIObeWlogwe can assume thatAnN to In factifNe N F B E8suchthat NE BandµCB oEherefore NtAE BiA andµCBA 0 soN A e NAuN Au MA andAnCN iAI ol
Def8hecompletionof µ is themeasure it in ftgivenbyutAoNI µLAI foranydisjointAodeNeN
ExerciseShowthat it B doesnotdepend onthedecomposition B AoNchosentocomputeµCAI
B eArUNF AzuNz µCAN teCAd
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Prof it is theuniqueextensionofµ to87DemoLetA eod Ne N and B ealbesuchthatNE BandµCBI o If ut is an extensionofµ tofee then
µCAI µ CAIk utAUNl k utCAoB µ IAoB EE µ lAlt µ B µ A
So utCAuNkettAoNI µ lAl 8hisprovesuniquenessso we are lefttoprovethat it is indeed a measureLet AioNn i ie IN a disjointfamilyofsets in ItZhen
et UfANNihet f il u live.atiD µ live.fiiEwMCAiti.InuTAiuNil
Purge CA ft et is a completemeasurespaceDenyFake B EEtsuchthat it CBS o andMEBWewanttoprovethatMeat But B AuN
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withAeGeandNe N Fherfore Ne BwithBtechandµCB'l O Eherefore ME AoBEA n lAoB
tk oandMe N SoMEST as we wantedtoproveObeseToheproofabovealsoshowsthatthesetN isalsothenullsetof it