...

() ... '-r' 'lJ7'

by user

on
Category: Documents
1

views

Report

Comments

Transcript

() ... '-r' 'lJ7'
()
AM..
,.
'-r'
,
'lJ7' "-'-';;';'«'
".'.
,>' . ."
!:~~~.!ys~~~!.t~!:~.tli~f;_,;~~~~
'fhrougiloUl
:IH~ (~Y,"'I"".
A\-l'iU
denc,te
the
r.eal
to A ~dll ho"'c...~r be <1t:~notedb:; ! f(x)ch:.
fIItr;,I:il.H.~).
In[e~1":\l::. v.i th respect
I
Lebe~g\jr..,! r."('~n:::\H'eora 11 (th~~
lit'
"Qh3~rable will ~lvays mean Lebesgua mcasur9ble.
f(t)dt.
1) Gtv~ ~xa~plcs af
~) J. seqUf!nc.~
:
(En)
of measutc;ble
suhsEts
'cf Rsothe;t
Hi
".;:J
EZ
:')
...
and
II>
~.( n En )
Hm AU:n );<
n 4'»
t)
1h.l
A contlm.lou:::
fum:t1on
on R: \/hi.:-h i~ flot un:! f.->rQ;ly.;:ont!r.uoI.1D.
. ::) A me..sIJrable f:mc:tlon ~n IO,1J, \lldeh 13 in tl{O,11:
d ) A .5eCIl lJcnC~
t of mc..surahle
func t ion$ \10 [O,ll\fi.th
II
"
vith
n ;):)t cot\VC'!rgillg' t:) zero a.e..,
$equ':!n('~~'t: n orm~'!asursble
func.tions
.
but. net
f(f n,(x)1
11\ L?I(I,~!.',
d:.: -j. O. lJ'.i\
f
,
'~)""
on [Or1)
'.,:Hb f...'+O
fl.e.,
.l
hut
,:lth
ILf r. ex}1 dx not convergingtb zero.
!) An (,'P'.Wd~I1.:;~ ~Hlh$(:t E 0 r R vi t,h ).( E) S 1.
2) ~:' State
thc"' 1)cmjnaterl
b J r:or.~p:J t r.
2
CO:)v~r.gEmce, theorem.
2
.~<=a )~ -rt
Yo
1Jm
J
-,,"--r
e-'
dx
11-~
0
x" H'"
'
J~~ti!y th~ step~ Joyour
J) R) Define
b)
$bo\(
conver~ence
th.tt
II. f n I
.~
calcul~t!onsl
in rucasure.
D l;nplics
f n ~'(I in nu.::1sncc.
it) J.et E bt. a m"?i'lsurabl..~ St)b(~t fjf nann aS51J:tle th"r. !'i\(f.) , <». L~t f n ,md 1. be
liip.aS~I,.nl"te r,mct!oo$
on f. ~.n~ B$:SlIme th~t
i) fn (x) 4 f(x) a.e. on ~
ii) fv:: all r. > 0 there exists
,
I
f
Preve
IfCI(t)ldt
that
<t
for ~ll
'f" I f n -f' l dt
'"a;.
a 0 > 0 such
~.
"' (I :.1S £14
0:0.
tltat. X(F) 0( 10.lr.inUc:-: i'hat
-.' '-..
5) De1\ I ~
i
'
f(x).. t
iix..O
O,~
1
xL sln
Sb)~
f
13
continuity
>: 6 (0,1}
of.9
funct!on.
absolutely continuous en [O,l}
c) (I~:ine vh.~n n ftmr.t1on
d)
if
:<
u) De~ine &bgolute
b)
-~
Zh.,'J f 113 Upschit7.
{fi)
0::
RlE!a::;u~::! 011 the
1,'(.~
on (0,1]:if.
7)
1.
defin~
~ j~ defined fo~ ull .c~sut~~le.~et~ B nnd \hat U i~ e
::r-~16.:i'ra
of
me.3suraule
a)
She"
tht\c
tJ i~
b)
:::ho/
that
(\.1~ ail}' opr:.:n interyu.l
c) r~n'3 the
< 2.
.\({ t) : t ~ 5)) .
Yo~ mDY a$su~e thRt
U{(~':h»
~
is L1psdd tz.
6) For a!y mQa~urcble ~ubset B of to,t}
J.I
1£
tthsoluteJy
sets.
c:.l)lItluu,'\I~
(a,bj
vlth
!.'i!$pect
c. jOt 1.1 VI:: },
t~) r"
,,:
rh
'1
.. J.a 3x" llx
the Ii.s.dors..Nikodyr.\ael'ivathr\!:
of ~..!",j tt. re~rif',,~i: to ~.
L.::tEl:c
.a Lt1bt:':s~l1efil('::asu[abl~
sub~et
of 10,lj;40,1].
~1(~CIlHeby Ex the
ly: {x,.,)G:
E} and by £'/ the ~et {x: (~:)y) .c:~Ej, rr(lVI~ (":t dl.~fH"O"P.:
':
\.
'
.,
..,'
1"
, ,.
)
1/
!
I. ",(EX' ~
H) (1) Let
f
.2 a. (l.,
~ l,l({:\.bD
lIen
AI,(y:
...(t..) r- 1.I! S .1 ~L.
I:
s\lch that
f{t)
dt."
0 f,.\!' ~1.1 ;( ~. (""hI.
that 1 ... 0 a.c.
h)
L::.l [ If: cl bt)lm(\;:.d fr.e8sf..lrabl~
tonction
"HI
-Hi
r
f(t)
c1t
r.
0
~I).
f(;(
a.ll
;-1'. f'1:-Cr'/e that
t(x.~l)
'" f(:()
:.i.P..
" t
P..
ami assm~~. that
:3};o)li
~~~t
Fly UP