ConstructionoftheLebesgueMeasure

Our aim is toconstructtheLebesgue meosureusingEarathiodory's EharemWe only havetoverifythehypothesisofthetheorem thatis weonlyneedtoconstruct a o additivefunction v onan algebraofsubsetsofIRwhichcoincideswiththeusualnotionofbrightforintervalsWestart observingthattheclass5 i Id IR Cab Catoo Cco bI j a eb E IR

is a semi algebra Wedefineµ 8 Eo too as

µ Colt O MkabD b a

µ IR µ CatooD µ C co bI I too f a eb e IRWe start observingthat µ extends in a uniqueway to a function w At5 lo tadWewanttoprove thot u is a additive But then it is

enough to prove that miso additive Ehedifferenee is that we onlyneed toprovethatif A is an intervalandAiearedisjointinterrrelssuchthat A Effi then uht.IE HailNoticethat foranyoneIN Az Ai andthereforeucNZFE.nlAil

fromwhereUCA iEµvCAi Weonlyneedtoprovetheconverse inequality First we assumethatA isofthefomca.b3 Eohentiisalsoofthefoomcai.b.itFheideaistousetheopencoveringtheoremtoselectafinitecollectionofAi'sthatalmostoverlaidBut Caidisnotcompactandalsotheintervolscambilarenotopen Letus get an e of room doingthefollowinga we defineAE fatqb limplicitezeb a

FhesetAE is compactAEeAand w lAl t v lAEl ezb Wedefine Ai Caibit itFhesetsAi are open AiE AEand

eEnvCAiI E iEnutAit Ez

Now AE E A EithAi e fi andthereforeAE isan open coverofthe compactsetAEWeconcludethatAEhas a finitesubcover CAIi j e J Nowwe can use additivitytoconcludethatw lAl e uFAd tg ejeg v t E tE

Ej gCufAp t

z Ez EIff vCAj t 2 Ez

t.ge4uwCAj tEzEEzEheextension to infiniteintervals is notveryenlightening so we leave it

ObsiZheopen cover theoremis an importanttopologicalpropertyoftheenvironment sets Althoughmeasure theory can beconstructed in arbitraryspaces thetheory ismuchmore powerful if weassume that D has nice topologicalpropertieslikeseparabilityandmetrizability

OpencoveringtoumLet K be compact and let CAi j i e I l afamilyofopensetssuchthat K E Ai Zhenthere is a finiteset 3 e I suck that

KE.Yej Aj

ExampleHausdorff measureNoticethat in Earathiodory'sEheorem thealgebraA is usedonly toconstruct an outermeasure

Eoratheodoujs8heorem.irLetutePCDdbeanouter measure Zhentherestrictionof a tothe r algebraofmeasurablesets is a measureReminder A is measurable ifµ B µ Bi A ME Bn A ABED

LemmeLet be a classofsubsetsof A Assumethat fee andthatshes a countable coverin E Let f to.eadbeseechthatfeels 0Zhen

MYA ifeng.ngfn.TL f Ai

is an outer measureLetustakeahelRd E Blair nerdino

fpCB.cn D VolCBcnrDNd ccdPdrP pedEj Bcn r nerd Oer'Er

MEp.ro AI iIyaiI4ufpCBiiner0

Noticethatuffprisanoutermeasure Zhen

U pCAIi fjnfM prCA

is also an outermeasure.Ehemeasurellapobtainedfrom EaratherodoujsZheoumis

calledtheHausdorff measure ofdimensionp

Def An outer measure is calledmetric ifutCAu B µ CA µ B whenever dIA B O

dCA B 0

Obey 8heHausdorff measure is metric

8hr1 Opensets are measurablewithrespecttometricouter measures

ObeZoheHausdorff measure UHp giveszero measure tosetsofdimension lapandinfinite measure to setsofdimension bp