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MeasureTheory
V.Liskevich
1998
1Introduction
Wealwaysdenoteby X our universe ,i.e.allthesetsweshallconsideraresubsetsof X .
Recallsomestandardnotation.2 X everywheredenotesthesetofallsubsetsofagiven
set X .If A\B =?thenweoftenwrite AtB ratherthan A[B ,tounderlinethe
disjointness.Thecomplement(in X )ofaset A isdenotedby A c .By A4B the symmetric
di®erence of A and B isdenoted,i.e. A4B =( A\B ) [ ( B\A ).Letters i,j,k always
denotepositiveintegers.Thesign¹isusedforrestrictionofafunction(operatoretc.)to
asubset(subspace).
1.1TheRiemannintegral
RecallhowtoconstructtheRiemannianintegral.Let f :[ a,b ] ! R . Considerapartition
¼ of[ a,b ]:
a = x 0 <x 1 <x 2 <...<x n− 1 <x n = b
andset¢ x k = x k +1 −x k ,|¼| =max { ¢ x k : k =0 , 1 ,...,n− 1 } , m k =inf {f ( x ): x2
[ x k ,x k +1 ] },M k =sup {f ( x ): x2 [ x k ,x k +1 ] }. DefinetheupperandlowerRiemann—
Darbouxsums
n− 1 X
n− 1 X
s ( f,¼ )=
m k ¢ x k , ¯ s ( f,¼ )=
M k ¢ x k .
k =0
k =0
Onecanshow(theDarbouxtheorem)thatthefollowinglimitsexist
Z b
|¼|! 0 s ( f,¼ )=sup ¼ s ( f,¼ )=
fdx
a
Z b
|¼|! 0 ¯ s ( f,¼ )=inf
¼ ¯ s ( f,¼ )=
fdx.
a
1
lim
lim
390103120.002.png
Clearly,
Z b
Z b
s ( f,¼ ) ·
fdx·
fdx· ¯ s ( f,¼ )
a
a
foranypartition ¼ .
Thefunction f issaidtobeRiemannintegrableon[ a,b ]iftheupperandlowerintegrals
areequal.ThecommonvalueiscalledRiemannintegralof f on[ a,b ].
Thefunctionscannothavealargesetofpointsofdiscontinuity.Morepresicelythis
willbestatedfurther.
1.2TheLebesgueintegral
Itallowstointegratefunctionsfromamuchmoregeneralclass.First,consideravery
usefulexample.For f,g2C [ a,b ],twocontinuousfunctionsonthesegment[ a,b ]= {x2
R: a 6 x 6 b} put
½ 1 ( f,g )=max
a 6 x 6 b |f ( x ) −g ( x ) |,
Z b
½ 2 ( f,g )=
|f ( x ) −g ( x ) | d x.
a
Then( C [ a,b ] ,½ 1 )isacompletemetricspace,when( C [ a,b ] ,½ 2 )isnot.Toprovethelatter
statement,considerafamilyoffunctions {' n } 1 n =1 asdrawnonFig.1.ThisisaCauchy
sequencewithrespectto ½ 2 .However,thelimitdoesnotbelongto C [ a,b ].
2
390103120.003.png
6
¯¯ L
¯
L
¯
L
¯
L
¯
L
¯
L
¯
L
¯
L
¯
LL
-
− 1 2
− 1 2 + 1 n
2 − 1 n
1
2
Figure1:Thefunction ' n .
2SystemsofSets
Definition2.1Aringofsets isanon-emptysubsetin 2 X whichisclosedwithrespect
totheoperations[and\.
Proposition.LetKbearingofsets.Then? 2 K.
Proof. SinceK 6 =?,thereexists A2 K.SinceKcontainsthedi®erenceofeverytwo
itselements,onehas A\A =? 2 K.¥
Examples.
1.ThetwoextremecasesareK= { ? } andK=2 X .
2.Let X =RanddenotebyKallfiniteunionsofsemi-segments[ a,b ).
Definition2.2Asemi-ring isacollectionofsets P ½ 2 X withthefollowingproperties:
1.IfA,B2 P thenA\B2 P ;
3
1
390103120.004.png 390103120.005.png 390103120.001.png
2.ForeveryA,B2 P thereexistsafinitedisjointcollection ( C j ) j =1 , 2 ,...,nof
sets(i.e.C i \C j =? ifi6 = j)suchthat
A\B =
n G
C j .
j =1
Example.Let X =R,thenthesetofallsemi-segments,[ a,b ),formsasemi-ring.
Definition2.3Analgebra(ofsets) isaringofsetscontainingX2 2 X .
Examples.
1. { ? ,X} and2 X arethetwoextremecases(notethattheyaredi®erentfromthe
correspondingcasesforringsofsets).
2.Let X =[ a,b )beafixedintervalonR.Thenthesystemoffiniteunionsofsubin-
tervals[ ®,¯ ) ½ [ a,b )formsanalgebra.
3.Thesystemofallboundedsubsetsoftherealaxisisaring( notanalgebra ).
Remark.Aisalgebraif(i) A,B2 A= )A[B2 A,(ii) A2 A= )A c 2 A.
Indeed,1) A\B =( A c [B c ) c ;2) A\B = A\B c .
Definition2.4A ¾ -ring(a ¾ -algebra) isaring(analgebra)ofsetswhichisclosedwith
respecttoallcountableunions.
Definition2.5Aring(analgebra,a ¾ -algebra)ofsets , K(U) generatedbyacollection
ofsets U ½ 2 X istheminimalring(algebra,¾-algebra)ofsetscontaining U .
Inotherwords,itistheintersectionofallrings(algebras, ¾ -algebras)ofsetscontaining
U.
4
3Measures
Let X beaset,Aanalgebraon X .
Definition3.1 Afunctionµ: A −! R + [{1}iscalleda measure if
1.µ ( A )>0 foranyA2 A andµ (?)=0 ;
2.if ( A i ) i >1 isadisjointfamilyofsetsin A (A i \A j =? foranyi6 = j)suchthat
1 G
1 X
µ (
A i )=
µ ( A i ) .
i =1
i =1
Thelatterimportantproperty,iscalled countableadditivity or ¾-additivity ofthemeasure
µ .
Letusstatenowsomeelementarypropertiesofameasure.Belowtilltheendofthis
sectionAisanalgebraofsetsand µ isameasureonit.
1.(Monotonicityof µ )If A,B2 Aand B½A then µ ( B )6 µ ( A ).
Proof.A =( A\B ) tB impliesthat
µ ( A )= µ ( A\B )+ µ ( B ) .
Since µ ( A\B ) ¸ 0itfollowsthat µ ( A ) ¸µ ( B ).
2.(Subtractivityof µ ).If A,B2 Aand B½A and µ ( B ) <1 then µ ( A\B )=
µ ( A ) −µ ( B ).
Proof. In1)weprovedthat
µ ( A )= µ ( A\B )+ µ ( B ) .
If µ ( B ) <1 then
µ ( A ) −µ ( B )= µ ( A\B ) .
3.If A,B2 Aand µ ( A\B ) <1 then µ ( A[B )= µ ( A )+ µ ( B ) −µ ( A\B ) .
Proof.A\B½A,A\B½B ,therefore
A[B =( A\ ( A\B )) tB.
Since µ ( A\B ) <1 ,onehas
µ ( A[B )=( µ ( A ) −µ ( A\B ))+ µ ( B ) .
5
F 1 i =1 A i 2 A ,then

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