Measure Extension Theorem

( r.

A,µ ) premeasure space .

Mt : 2h → Ceos ] Carath'eeday outer measure

MY E) = in f { ¥, MIA;) : Aj EA , Es Aj}Theorem ( Frechet

,Carathieodory , Hopf, Kolmogorov, . . . 1920 s)

There is a o - field MIA sit. film is a measure .

Ti. MA) SM

j

standard approach : M -

- { Esr -

- MKT) --tiltE) tutto EY, ftsr}↳how it is a o - field

, containing A . Requires new tool :• Show µ* is countably additive on it } Monotone Class Theorem

or

Dynkin 's D-X Theorem

Advantage : works for all premeasures .

Disadvantage : finicky , technical , unmotivated : too clever by half .

river's Approach restrict to finite premeasures .

(r,A,µ ) µ

: A → co,Mohl

want extension : pi :'

I /1

.

Make 2A into a topological space2- Define it to be the closure of A- .

3.

Prove µ-

- A → ↳as) is sufficiently continuous ,i. extends to closure

.

4. Use topological tools to show it is a o- field

,

and it is a measure .

It will turn out that it .-pi on A

Pseudo - Metric Spacesd : Xxx → Egos)

1.

dlx, y )

-

- o ⇐M=yEF 'n R2

,d l 1471) : toe ,- y , I

2.

d cosy ) = dug ,x )

3. dlosz) s d logy ) tdly , z)

- A sequence cardie , in X has a limit a fTe >o F NEN Fn> N d can

,

K ) < E.

. Given VEX,the closure T is the set of limits of sequences in V .

.

. A set V is closed if I =V.

. A function f : V → IR is Lipschitz if 3 Keyes) s - t. lfcxj - fly) Is KellyD .

Prof : If f is Lipschitz on a nonempty Vs X , then there is a uniqueLipschitz extension f : T→ IR cwith the same Lipschitz constant K ) .

fly = f .

The outer Pseudo - Metric

( r,

A,µ ) finite proneasure space .

put :P → Centro carattieday outer measure

MY E) = inf { ¥, MIA;) : Aj EA , Es Aj}Def : dm : the → Co

, you ]

dm ( E,

F) =

µ*LEAF )

EDF - felt OCFIEJ

E F

Prop . die, is a pseudo - metric on 2?Bf . o

.

Takes values in CymruV

t. du LE,E) EMCEE

E) =p*lol ) -- o

2.

Ed F = FAE3

. Triangle inequality - HW. %

key Properties of the outer Pseudo - Metrict.V-A.BE 2h data,B)

'

- du IA:B' )

.

2. TIME , ,{ Bdf, t 2b

Cas dal An Bn ) s ⇐ dylan,Brdlb de An

,BD s dawn, BD .

Pf . 1.

A'a B'= H4B9olB4A9=# dB lol Bee,

') = IBN 01AM)= BoA

i. dm ( A:B'

)=pMAkB9= -MMBAA)

= d.nl BA) = dying B) .

2. ca) ( LEAD a BD s ESCANABA LAW )

i - dal E.An.nu?.BD--rM"744 Is ?EmMAraBD

-

- Eiden, BD

Lemma If 15h, A,µ) is a finite promeasure space ,and Ant A with Ant A

,then dalton,A) =p -MAD→o .

Pf . Let Dn -- Ant Ant .

Then A - Dn,

and by definition ofµMHA) s MDD = ftp.i?plDD=fnzmnu!Dnffnjn.y1ADAn GA i. MAN -MAD SMH SMHA)

-

: finalAD =p * CA) .

Fix n.

Note that Anton P Alan .

So repeating t

i - ma l MAD = Kyzyl ANAD = finna fu IAD -y IAD)" P = MMA ) -MAD ⇒ 0

.

didA,Antti't Aa AD H

Cor. If Ano- A and Ant A

,then AG A-

.

Theorem . If he, Apu ) is a finite premeasure space ,

then the closure A- of the field A in the

pseudo -metric space 127dm is a o - field .

If . t .

¢ c- As I v

2.

Let Bo- A-.

i.

7 An c. A s.

t.

Arndt B.

.

'

-

du l Ai , BD= dylan, B)→

o.

i . An'→ B'

.

q i ."

B.

3.

let 1BDii , G- A-.

Fix e > o.

For each n,find some Ant A

H- du IA n, BD

d Etz.

-

'

- dal Ean , I, BD s E. dawn,BD E En - e .

Iste i . For each m,

find A-me to sit .

of Him, Busts .: indene Bnft i

. HI