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

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

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

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

withAeGeandNe N Fherfore Ne BwithBtechandµCB'l O Eherefore ME AoBEA n lAoB

tk oandMe N SoMEST as we wantedtoproveObeseToheproofabovealsoshowsthatthesetN isalsothenullsetof it