Lebesgue Measure ( § 6.6 in Driver)he Radon measure on IR satisfying

Ika,bD = b- a,

- es -as bees

is called Lebesgue measure .

It is the most importantmeasure on IR

.

Notice : if a ER ,J -- la

,ble du

,

then

XCJTTI = Xdatt ,bttl ) = Lbt- Latif -- b-a -

-NJ)

.

⇒ X ( Att ) = ALA) for AG Bcg HR)⇒ lil Ett) = ME) for ad E SIR [ HW 3)

Theorem : X is the unique translation invariantBorel measure st

. 716.1151 ; if µ is

another translation invariant Borel measure ,then µ = a X fer some a > o ,

Pf . Since is translation invariant,

A - Mazar, is too ,and all 1) = t - 0=1

.

For the converse : S ' pose µ 16,111=1 , µ is trans.

inv,

o'a Z En

telo.lt ' II ( ka ,

in = EI ( Etcs 's ) )i.

night 's . finer -

- Eigenstate )Eg

' n'messI

✓

mle.mg/--mn-.m,nEPFnlm-nI-Fnld--mnV-rEQFnlr) - Face) =r .

⇒ i. Fmla )- taek i. µ=X .

Engel -- af-

-o, mum - MILE.de,

Mtn) ) - Ease -

- O.

> o ,i . £µ is trans . inv - ad # give f. any i. It

By similar reasoning : if SEIR,BE BARS

,

X ( 8-B) = 181 - X IB) f - B -

- { f. x .- KE B}Pf . If 8=0

,tx ( go}) -

- o-

- lol - WB)

If Ho, µ IB ):-

- Fg, HE. B) ←This is amethyst : MV -(HD

i. vk.es -

- f , Neo - 6,11)↳ see

,Heil = 8 → me. H

-

- I

↳ Ko, MyoD -

- AHHHH, o)-

- a talked - Bok= o - I = 181

i. M

-

- X .

H

Null sets ( § 6. le in DriverIn a measure space trip) , a measurable set NEF is

a null set if µ ( N )-

-o -

Eg .

If µ= 8 .. , anyset not containing x. is null .

Lebesgue null sets :

e. If Xo IR is countable,then XCX) - o

.

{ n'hype, . .. } X s Gifs , any forany e >o

.

"HEEHaig , xp .

-E. Ene .

" HH -

-o.

• There are lots of uncountable null sets,too !

en sa wc . X Koff of 2n internalsEL XL G) ' 213 of length Ign .

•: x LG) -49

C - Gen i Ngs est Ase . ⇒ a' ⑤n

Problem : most subsets of the Cantor set are not Borel sets .

That is : there are many sets NE BIR ) of Lebesgue measure othat contain non - Borel sets A EN

.

This can sometimes causetechnical problems .

Definition : A measure space 038,µ) is called (null ) completeif,for every N E F with µ IN)

-

-o, every subset

As N is in I land i. µ I A) - o ) .

Theorem For any measure space Cross, µ),there is an

extension I 28, it IF =p , s t . 058,9 ) is

Pf . The most obvious thing actually works :

§= { AVA : AGE,her, FNEJ-s.t.MN) of ASN }

Tel Aod) : - MCA ) .

.Eisa G - field containing 8 : FSE i. of

,ref

.

. (Aug )'- A' off = (A' n DINA

'

Nc ) GE.

T W

(Nlt) ON'

n tf

. # plan 0AM = YANO Onan mhfndfn.IM#lNnD=o .

§ In FF '

Ganefai. C- g-

F- = { AVA : AGE,her, FNEJ-s.t.MN) of ASN }

Tel Aod) : - MLA ) .

• it is well-defined : suppose A. A 's d. fit 't , no-

- old 's -- oin

Aux = A' ON .

[A E Ault c. AMUN

'= A' u

'= A' UN

'

.

i. MIA) split'od ') s MA

'

It - M&A,canopy

← MIA's sand quad. A is a measure

.

Ano An disjoint i. Alun Anon) n

§ In mHnI⇒.

-

- my ↳ Anu 4AD =MGdADTa subsetof U Nn

is a ndllstt

. (r,Fyi ) is (null ) complete (by construction) .

The Lebesgue o - Field.

often,when one speaks of Lebesgue measure , one implies a particular

large o - field :

M : = { E SIR : FA SIR WAH AnE) tilAnEc )}

↳ BUR) § M

↳ M is null complete land bigger than Btr )we will never use this M . For us

, Lebesgue measureis a Borel measure ; worst case, we may need to complete theBonet o - field in some applications .

BTW, fun fact :

In a finite Lp re )measure space 03A,µ ) ,

Mm = A- I the closure in the pseudo - metric dm )[ Driver

, Prop . 7.111