IR We ,zfdx Cr - University of California, San Diegotkemp/280A/10.2.Lebesgue-vs-Riemann-After.… · Lebesgue us Riemann Let IIR B, 7) be the Lebesgue measure on Borel sets in IR

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}

[email protected] II,

- 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