Properties of Measures(1.3)_

definitions:asetX, M C pox)a a-algebra_ {EEMINEM,

3 E;CM

•1) FEMman

ï a measure-on ox,M)is a fn.N: Z]2.503 st. _ ^1 uld)=0

2.is. E;CM disjointD

DE;) = { white. j=1ï

NEEi)

"countable additively".(X,Mif)is a measure space_

I "measurable sets"

Thn in(HMF)measure space.

(a)EM 'EGF * NEE) "monotonically"

(b)E;ElitismH NC is] 5 5TEE;)E(1countable subadditivity"

(c)Egoism,E,ïEco. F. white.to

E,)~CEi)- him 350(d)room, Feb. 5,Ego * ï (NE.) = him j=is 55 C NE;)

Riordan Not)(0}J"continuityfrom above, below'\

rumeno

Proof:(a)

o{F o E111,

F -F E u _ - | \E (F\E:= FrE')

f disjoint

/ / 11111 1111ino A|\

M (algebra)

=) (prop. 2.) NE) =N(E)+fF\E)INEv

(b)exercise text(c)

OE,otoE,ois.

o U

aof •5L;= E,U (ENE,)UTE,\Ez)u 000

Ä huh disjoint,EM

L(prop.2) 8, = (•1 NIf,

to

NEEj\Ej-1)Y

1200Ei

= himNEE;\E;: )o Eu

=limwant Engvo.

o'

(d)experiment.Why is n (E.) CD needed?Ex:E j =jiao)( c IRl if,NE,=dNE canting measureNLEi)=o,fl0)=011 _ _

more definitions:measure space XML)(I is finiteif Nixonm is offsite if _FEE;' bitCm,Nationst.4=8. it;_

P.

:Nom is- null

re ofN(N7=03 measure space iscomplete ifM includesall subsets of null sits

Rem _:any measure sp.can be completed"1/ in N natural wayThe. T..)

x

next: toward lebesyve measure.IR ( ](161111

a b| |de ?111111111]action

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length. bra,~ bear~

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