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Applications
49
simplefunctionisstronglymeasurable.(Letusalsonotethataswith
scalarfunctionsandLebesguemeasurability,continuousfunctions
definede.g.onintervalsofthereallinearestronglymeasurablewith
respecttothe
σ
-algebraofeitherLebesgueorBorelmeasurablesets.)
WecannowdefinetheBochnerintegralitself.Forasimplefunction,
thesituationis,well,simple:
∫
X(
Σ
i=1
n
yiχA
i)du:=
Σ
i=1
n
u(Ai)yi.
(Itcanbeprovedthatthesumontheright-handsideiswell-defined,
i.e.,thevalueofthesumdoesnotdependonthechoiceoftherep-
resentationofthesimplefunction.)Astronglymeasurablefunction
f:X→E
iscalledBochnerintegrable,ifthereexistsasequence
(fn)n∈N
ofsimplefunctionssuchthatthecondition
n→∞∫
lim
X
"fnlf"du=0
issatisfied.(TheintegralintheaboveformulaistheLebesgueinte-
gral.)Ifthisisthecase,theBochnerintegralof
f
,denotedby
∫
Xfdu
,is
definedasthelimit
n→∞
lim
∫
Xfndu
.(Ofcourse,itisanelementof
E
.)
Itcanbeprovedthatthislimitdoesnotdependonthechoiceofthe
sequence
(fn)n∈N
.Further,theBochnerintegralofafunction
f
over
somemeasurableset
A∈X
canbedefinedas
∫
Afdu:=∫
XfχAdu
.
(Equivalently,wecoulddefinesuchanintegraltreating
A
asthemea-
suresubspaceof
X
.)Itcanbeshownthatif
f
isBochnerintegrable
onX,thenitisBochnerintegrableonanymeasurablesubsetofX.
ThepropertiesoftheBochnerintegralarequitesimilartotheones
oftheLebesgueintegral.(Infact,forfunctionswithvaluesin
R
both
notionscoincide.)Inparticular,theBochnerintegralislinearwith
respecttotheintegratedfunctionandadditivewithrespecttothe
setofintegration.Also,continuousfunctionsareBochnerintegrable
(again,wecanusethe
σ
-algebraofLebesgueorBorelmeasurablesets
here).Further,thenormoftheintegralofaBochnerintegrablefunction
doesnotexceedtheLebesgueintegralofthenormofthatfunction
(whichdoesexist),i.e.,
"∫
Afdu"<∫
A"f"du
foranymeasurable
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