Want to extend µ to a measure ti -- off) → God .
Ie .
Find a o - field I 261A ) and a measure fu on 8 set
- felt =p .
.
The extension is unique if µ , and i. it , is
o - finite .
Caratheodorg's Extension
Let g :E → egos ] s
.t.pk/I--o .
pHA) = inf { E. HE;) : Eje E , As Ej }
A* "
Theorem : If E is a field and g is a
premeasure , then g " lo CE)
is a measure .
. pilot 1=01 2
V
/
2 .
" if "if " ,
E.fl tflAnd + En t .
-
s fellow t En H
Def Let r be a nonempty set . A function W : ER→ Co
, as ]
1 .
3 .
An) s VLAD .
Thus ,
,
outer measure .
We can use it ( in theory) to distinguish finitely - additive measures from premeasures .
Lemma : If th , A ,µ) is a promeasure space , and
(both ,V ) is a measure space extending it , then
v spit on GLA ) .
.
- - } = µHB) . H
Prep : If th,A.X) is a finitely - additive measure space, the x " Ex on A
, and X*=X on A iff X is a premeasure .
Pf .
-XD I
A sun An .
Then A ' UnlAnoAl Bi - B , ,Bn " Bull Bio - - - oBuds Bn B. ⇐A
= YB "
• If Xi --X on A , then X is a premeasure .
let An c- A st .
A -- in ,
= EzErland s x to
⇒ ⇐ Xiao = xD H