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
Recommended