outer Pseudo -Metric closure ( 86-2 in Driver ) www.ylanpd. ( r

,Apu) finite promeasure space -

-MAD• ME) = inf { Earl : Aj GA , Is A ;) FEET. dyce,F) = fittest)Theorem : he closure Tt of A in the pseudo - t

metric space ( 27dm) is ao - field .

Now,we've proved that Mala -µ .

So,

for A.BEA ,A

B

da CA,B) =µ*lAq,

B) =µlAaB)A =p CHUB)

- MLA'B) 31mA) -MBTI

FIB) - Fuld i. Me Lip- I on A

MLA )- MIB)

Prep µ extends to a unique ⇐Lip - t function it : b-→ ↳ pond .

Def . Given E S2? Es :-

- { countable unions of elements of E }↳ Note : Es is automatically closed under countable unions .

If E is closed under finite intersections , so is Es :

Eid ''D= if.ij .

Restatement of Lemma (from last time) :If Ed is a finite promeasure space , then AT =D , and it =p

"on Ao .

Pf . We showed that if A o An f A then dylan,A)-MMA) -MAD If .

If AE Ao.

i.

A - BanA

' AE Ae A- .

⇒

A- SAI SAID

⇒ a.pay

data, so-so

Define Are Bi to - to

MHA)-

- LIMAD -tilt) . x

Prop .Let Ir

,A,µ) be a finite promeasure space .

For Be 29 TFAE :

ID BE It.

(2) te > o,FC e- As St

.

Bsc and pilesB) =D, I B. c) a E .

Pf.

(2) ¥112 : select a sequence Cnt Ao s.

t. did B.Cn) s t ; then Cn→ B

and so BE AT = To.

(1) ⇒ 12 ) : Fix e > o. By CD , And Get

Suk e > dm ( B,

G) =p* (Ba G)

E >MY Beak inff plant. AnEA, B

-G s An}

i.

we can fnd HI , in Asot

.

Bags ⇐ An, ⇐MADEset c.= Go An c- A-

. & B s (Big) OG s LBAG) UGSC

and QB -

- CGI B) u An IB) s (BAG v AnIB) s An

i. pilot B) SEIMAD - E . µ

Cor : Let Ir,Apu) be a finite promoas are space .

Then µ -

- fi on To .

Pf . Let Beto . picB) = Igi IBI- fields delB,d) =p'MBad) yuhB)

For reverse Meg: Fix ere

.

Cheese CGA, s

.

t.

BGC ad

Hml Bc) ; m*CAB) s E .

Itil BI -plateful B. C) < e€.,

'" grimeFCB) MHC) spice) SpitBITE

Take Ebo : MLB) TMB)#

Theorem : If Cr,A,y) is a finite promeasure space ,then it : to→ co

,mm is a measure .

Pf . We will show that pi is finitely - additive on A-.

Once we've done that : we've shown p is a finitely -additive measure on the o - field to

,and i

.

it is↳inn!!! as:P%edddn.it: ii.aiijbgubhaoeiiiino"ayy

> Let A.Bet . i. Find An→A,BPB

,An,

Bn GA .

day::: :3 's :b, is de"'m tdr"'m → " %,

i - Ju (AuB) qui (AaB) = fish fu (Ano Bn) xp I Ano BD)

= LEI fulAnd tu IBN =p lb) ti IB