Lecture 3

Recall Measureswhat kind of familyofsubsets should we

restrict to

Let X be a set

Eelgelia unonemptyCtn algebra is a collection of subsetsof X that is closed under fine unionsand complements countable

Q Suppose X is a topologicalspace is

A all open sets and closed sets an algebra

A X IR Lo Mu 1,23 4,2 is neither opennor closed Consequently CA is not closedunder finite unions so it's not an algebra

Instead our goal will be to find thesmallest o algebra containing all openan sets

X we will getthis for free

unonempty

3Given a family E ofsubsets of X the

algebra generated by E denoted MCEis the smallest o algebra containingE

that is all other algebras Tcontaining E must contain chCEl

Let CXd be a metric space

Def The Bord oalgetia of X denotedBx is the o algebra generated by theopen sets in X

often call sets in Borel algebrB ordsets

Countableintersection

what do Borel sets look likeopensels quoteLet's go from the inside ont t

a e.g Iefin 1 to DL open setsTo all countable unions of sets in FIS all countable intersections of setsinF

F complementsof sets in F

To build Borel algebraI Frs Is u uncountablemany Bn r

closed under closedunder no longer closedc inters complements under c intersections

Prep The Borel o algebra of IR denoted BRis generated by each of the followingCil EI Ca b al bIii Ez Eab iac b

Iiii E3 Cab a blin Ey Ca ta AERv Es Ca ta at IR

Moral You can get the whole Borel algebrastarting from these simple building blocks

PI HW 2

We know what BIR looks likeWhat does Bpd look like

We will consider the general case ofproduct o algebrasLet space algebraCe g Bpl

Xa Make A be a collection ofmeasurable spaces for A countable

X IfkEe gRd Ai Xi Xz Xd Xi

Its projectionof X onto Xa

Def the proft lgebra IQ.dkis the co algebra on X generated bythe sets Ita Ea Estella

e g rectangular I gCab e g BIRprismai bDxLacbDX CadBd

What is the relation between Bpa and BRT Tspoiler theyarethesame

Props Suppose Us is generated by Ea with XIEThen elk is generated IIe 2 Ea cGIsindefnabove we hadella here e g BR jnow we can restrictto smaller familye gCab

Recall if E E 2 then CMCE Ect Tnif E E IFT the n ell C E Edt Fr

Pf ByahUK IeEa Ea c EB Echl RHI Ea Estella

H

acMxIt remains to show opposite containment

By d it suffices to show

Ita Ea Estella EMELI 2 Ea EETEod

Note that LI Ex x EX Kahne Eats EAIea x EX Italy teaTea HI Ea

Since Eaa is a co algebra it suffices to showITI't Ed EEooo V ELE ella LEA

Because Xx C Ex and HIYED X xXzx Eaxif EL C Ex then Tia EL E 8000

y AI LEI

Tintin.ms Efii nII.IEisaiEaisewa E imust contain ell x l 4

Thus ITI't EN E Toooo for all Ea C ella 14I 19added detail afterclassd

We will now show B pd h BIR

In order to prove this it is convenient to

endow product space X Xi withmetrdmffx.mu

xe Ig ya yzD FIFed.fxi.yi

Forexample Xi d IR il l this is ldfemkx.xd.co e

I dsumkxyxd.cooD

dmaKxyxz co MEI

of course the Borelo algebra Birdonlydepends on the notion of open and closedsets and Imax is equivalent toother distances

dsumlfx.inzi ixdYlyi yz ydD idilxi yil

For example Ni d 4112,1 1 this is l

The Consider metric spaces X Xz Xdand X If Xi endowed with the productmetric drmax then 0 Bxi E Bx

a

I 1

If the Xi's are all separable Bxi Bx