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48
Measuresofnon-compactness
1
t∈[a
,
b]
thereisaconstant
Mt>
0suchthat
"x(t)"<Mt
for
every
x∈V
.This,togetherwiththeequicontinuity,impliesthat
V
is
boundedasasubsetof
C([a
,
b]
;
E)
.Indeed,let
U1
,
...
,
Un⊆[a
,
b]
be
afinitecollectionofsetsformingacoveringof
[a
,
b]
withthediameters
notgreaterthan
δ
,wherethenumber
δ
correspondsto
8:=
1asin
Definition1.2.4.Moreover,ineachset
Ui
letuschooseapoint
ti
.Then,
forevery
x∈V
and
t∈[a
,
b]
,wehave
"x(t)"<"x(t)lx(ti)"+
"x(ti)"<
1
+max
1<i<n
Mt
i
,asthepoint
t
belongstosomeset
Ui
.So,
"x"∞<
1
+max
1<i<n
Mt
i
for
x∈V
,whichmeansthattheset
V
is
boundedinC([a,b];E).
ToendtheproofitisenoughnowtoapplyAmbrosetti’slemma
andTheorem1.1.20(a).
1020100Remark0
Itisworthnotingthatnotonlydoestherelative
compactnessofanon-emptysubset
V
of
C([a
,
b]
;
E)
followfromits
equicontinuityandthefactthatthesections
V(t)
arerelativelycom-
pactin
E
,butitisinfactequivalenttothosetwoconditionscombined
(see[138,Theorem1.2.5]).
InthesequelwewillusethenotionoftheBochnerintegral,whichis
quitesimilartotheLebesgueintegral,butforvector-valuedfunctions.
AsinthecaseoftheLebesgueintegral,beforedefiningitweshallde-
finesomeauxiliarynotions.Let
(X
,
X
,
u)
beafinitemeasurespaceand
let
(E
,
"·")
beaBanachspace.Asimplefunctionfrom
X
to
E
isafunction
n
oftheform
xl→
Σ
yiχA
i(x)
forsomepoints
y1
,
...
,
yn∈E
andmea-
i=1
surablesets
A1
,
...
,
An∈X
.(Obviously,thisrepresentationofasimple
functionneednotbeunique,butthisdoesnotposeanyproblems.)
Afunction
f:X→E
iscalled
u
-stronglymeasurable(orjuststrongly
measurable,whenthereisnodangerofambiguity),ifthereexistsase-
quence
(fn)n∈N
ofsimplefunctionssuchthat
n→∞
lim
"fn(x)lf(x)"=
0
foralmostall
x∈X
.(Thisconceptisnotthesameasmeasurabilityas
definedinthePreliminarieschapter,althoughinthecaseof
X
being
aboundedintervalof
R
withtheLebesguemeasure,itis–asthename
suggests–strongerthanLebesguemeasurability.)Inparticular,every
