View
3
Download
0
Category
Preview:
Citation preview
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
Recommended