Measure Extension Theorem

( r.

A,µ ) premeasure space .

Mt : 2h → Ceos ] Carattieday outer measure

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

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

There is a o - field M IA et. film is a measure .

Standard approach : M :{Esr -

- MKT) --tiltE) tµ*lTnE'

),

Asr}↳how it is a o - field

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

or

Dynkin 's at 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,

es )

want extension : pi : 't I1

.

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 .-µ on A

Metric spacesd : Xxx → Egos)

1.

dla, y )

-

- o ⇐ a- y2.

d cosy ) = day ,x )

3.

dlx,z) s d logy ) tdly , z)

. A sequence cardie , in X has a limit x 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 premeasure space .

put :P → Centro carattieday outer measure

MY E) = in f { ¥, MlAj) : Aj EA , EE Aj)Def : dm : the → Co

, you ]

dm l E, F) = MHEst )

Prop .dµis a pseudo - metric on 2?

Bf .

key Properties of the outer Pseudo - Metrot.V-A.BE 2e du LA,B) = du IA:B

' ).

2. FIAT!,{ Bisi, t 2h

Cas dal An Bn ) s ⇐ dylan,Brdlb du An

,BD s dawn, BD .

Pf .

Lemma If th, A,µ) is a finite promeasure space ,and Ant A with An TA

,then da Hon,A) =pMAI -MAD→o .

Pf . Let Dn -- Ant Ant .

Then A = Dn,

and by definition of u*

MHA) s §,MIDD

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 .