IR We ,zfdx Cr - University of California, San...
Transcript of IR We ,zfdx Cr - University of California, San...
Lebesgue us.Riemann
Let IIR,
B,7) be the Lebesgue measure
on Borel sets in IR .
We now know how
to define f,zfdx for fell-
Cr,BIR)
.
Eg.
f = I cog ,← Q dell E BC IR)
-
'
.
G Lt
If di =/ Dance,odd= HaigGil )
-
- o.
can table
so we can integrate non - Riemann integrable things .
Thm : al .SI Let B denote the completion of BCR)wrt X
.
Then a bounded function f : Lgbt → IR isRiemann integrable iff it is BoB measurable
,
and X { xecgbl : f is discontinuous@ x ) -- o .
In this case, fast
,
DX = lift"'d"'
us. ,t;hI; II! -- titi ,Partial Proof : S -
pose f is Riemann int-bl 'd
, y ⇒ CI;)i. He> o can find it -- { a - Katie . -
- Ctn -b}
- tf Ee its# f. II; I '- fseodx
ffdx s-
- Seidl Ea-
- Io 'Eff dis , . Sfr
The Lebesgue integral also allows us to handlei. improper integrals " . Assume -130 GntmensEg . I;fiada:=fiFIf""ok
Dan,n ,stem,nt0
A =kFzf,jdX -
-
- flan,n, sfban-yni.isCs
Ff= fry. ffbc.n.n.dk LJMCT
feta) =
ftp.flsc.ynodt-ffdx.
L
what about other Bord measures on R ?So long as µ L -yn ] -
as the N,we mean
y FT i . F'30 .
Radon measures,and so µ -up i.e
. µ Lgbt-
- Fcb) - Fla) .
F is right - continuous . Suppose that F is d- (HD
Mla, bi = Fib)-Fla) = fab F'la) else .
= fam, F' db .
Define VIA) -- SAF'DX
. Then V --µ on do .
i. pepe an oldu) -- Blk) .
i. !! du-
- faff' ditfetilcgbs.ms#lflFfeL4ca.bsD--fabfixsFbddx
.
centrums.
of course,not every Radar measure has a density .
Eg . µ-
- 8.= ME with F = Dcps) .
µ (A)= f l f re Ae f xp A
= f,edt !f ca,bi xx
eat-
- o.
Me .
⇒ e-
-o a.e.ad
on la,bi
.
i. f-
- o a. e. HI on IRYX}
↳ old ⇒ . ⇐⇒ ee a.e. on an IR
we see that F had better be continuous ( it .Mp has no point mass )if we want Me to possess a density .
To mimic the calculations on the last page , we may not needFact, but we at least need the Fundamental Theorem ofcalculus to hold
In general : if F is continuous,dfferen liable a. e.
,
and nice enough thatFla) = I F
'
DX for A - a. e.
x
capo
then we can mini the preceding to see that
MF CA) = ↳ F'DXIf F e- Ct
,this is fine .
It works much more generally - butit doesn't always work , even if F is continuous
,and dffble a - e
.
Eg.
The Devil 's staircase F dff-ble a. e.,F'- O
.
But f F'dk = 0= Fix) the > o
.
Csx I