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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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