4. Measurability - Probability Tutorials

1 Tutorial 4: Measurability14. MeasurabilityDefinition 25 LetAandBbe two sets, andf:A Bbe a A,wecalldirect imageofA byfthe set denotedf(A ),anddefinedbyf(A )={f(x):x A }.Definition 26 LetAandBbe two sets, andf:A Bbe a B,wecallinverse imageofB byfthe set denotedf 1(B ), and defined byf 1(B )={x:x A, f(x) B }.Exercise two sets, andf:A Bbe a AandB Explain why the notationf 1(B ) is potentially Show that the inverse image ofB byfis in fact equal to thedirect image ofB byf Show that the direct image ofA byfis in fact equal to theinverse image ofA byf 4: Measurability2 Definition 27 Let( ,T)and(S,TS)be two topological spaces. Amapf: Sis said to becontinuousif and only if: B TS,f 1(B) TIn other words, if and only if the inverse image of any open set inSis an open set in.

2 We Writef:( ,T) (S,TS)is continuous, as a way of emphasizingthe two topologiesTandTSwith respect to whichfis 28 LetEbe a set. A mapd:E E [0,+ [is saidto be ametriconE, if and only if:(i) x, y E, d(x, y)=0 x=y(ii) x, y E, d(x, y)=d(y, x)(iii) x, y, z E, d(x, y) d(x, z)+d(z, y) 4: Measurability3 Definition 29 Ametric spaceis an ordered pair(E, d)whereEis a set, anddis a metric 30 Let(E, d)be a metric space. For al lx Eand >0,wedefinetheso-calledopen ballinE:B(x, ) ={y:y E, d(x, y)< }We callmetric topologyonE, associated withd,thetopologyTdEdefined by:TdE ={U E, x U, >0,B(x, ) U}Exercise the metric topology associated withd,where(E, d) is a metric Show thatTdEis indeed a topology Givenx Eand >0, show thatB(x, ) 4: Measurability4 Exercise that the usual topology onRis nothing but themetric topology associated withd(x, y)=|x y|.]]

3 Exercise (E, d)and(F, ) be two metric spaces. Show thatamapf:E Fis continuous, if and only if for allx Eand >0,there exists >0 such that for ally E:d(x, y)< (f(x),f(y))< Definition 31 Let( ,T)and(S,TS)be two topological spaces. Amapf: Sis said to be ahomeomorphism,ifandonlyiffisa continuous bijection, such thatf 1is also 32A topological space( ,T)is said to bemetrizable,if and only if there exists a metricdon , such that the associatedmetric topology coincides withT, = 4: Measurability5 Definition 33 Let(E, d)be a metric space andF metriconF,denotedd|F,therestrictionofthemetricdtoF F, |F=d|F (E, d)beametricspaceandF E. We defineTF=(TdE)|Fas the topology onFinduced by the metric topology F=Td|FFbe the metric topology onFassociated with theinduced metricd| Show thatTF T GivenA T F, show thatA=( x AB(x, x)) Ffor some x>0,x A,whereB(x, x) denotes the open ball Show thatT F 4: Measurability6 Theorem 12 Let(E, d)be a metric space andF ,thetopology onFinduced by the metric topology, is equal to the metrictopology onFassociated with the induced metric, (TdE)|F=Td| :R ] 1,1[ be the map defined by: x R, (x) =x|x|+11.

4 Show that [ 1,0[ is not open Show that [ 1,0[ is open in [ 1,1].3. Show that is a homeomorphism betweenRand ] 1,1[.4. Show that limx + (x) = 1 and limx (x)= R=[ ,+ ]=R { ,+ }.Let be 4: Measurability7as in exercise (6), and : R [ 1,1] be the map defined by: (x)= (x)ifx R1ifx=+ 1ifx= Define:T R ={U R, (U)isopenin[ 1,1]}1. Show that is a bijection from Rto [ 1,1], and let = Show thatT Ris a topology on Show that is a homeomorphism between Rand [ 1,1].4. Show that [ ,2[,]3,+ ],]3,+ [areopenin Show that if : R [ 1,1] is an arbitrary homeomorphism,thenU Ris open, if and only if (U)isopenin[ 1,1]. 4: Measurability8 Definition 34 Theusual topologyon Ris defined as:T R ={U R, (U)is open in[ 1,1]}where : R [ 1,1]is defined by ( )= 1, (+ )=1and: x R, (x) =x|x|+1 Exercise and be as in exercise (7).]]]]

5 Define:T =(T R)|R ={U R,U T R}1. Recall whyT is a topology Show that for allU R, (U R)= (U) ] 1,1[.3. Explain why ifU T R, (U R)isopenin] 1,1[.4. Show thatT TR, (the usual topology onR). 4: Measurability95. LetU TR. Show that (U)isopenin] 1,1[ and [ 1,1].6. Show thatTR T R7. Show thatTR=T , that the usual topology on Rinducesthe usual topology Show thatB(R)=B( R)|R={B R,B B( R)}Exercise : R R [0,+ [ be defined by: (x, y) R R,d(x, y)=| (x) (y)|where is an arbitrary homeomorphism from Rto [ 1,1].1. Show thatdis a metric on Show that ifU T R,then (U)isopenin[ 1,1] 4: Measurability103. Show that for allU T Randy (U), there exists >0suchthat: z [ 1,1],|z y|< z (U)4.]]

6 Show thatT R Td Show that for allU Td Randx U,thereis >0 such that: y R,| (x) (y)|< y U6. Show that for allU Td R, (U)isopenin[ 1,1].7. Show thatTd R T R8. Prove the following 13 The topological space( R,T R)is 4: Measurability11 Definition 35 Let( ,F)and(S, )be two measurable spaces. Amapf: Sis said to bemeasurablewith respect toFand ,ifand only if: B ,f 1(B) FWe Writef:( ,F) (S, )is measurable, as a way of emphasizingthe two -algebrasFand with respect to whichfis ( ,F)and(S, ) be two measurable spaces. LetS be a set andf: Sbe a map such thatf( ) S as the trace of onS , = |S .1. Show that for allB , we havef 1(B)=f 1(B S )2. Show thatf:( ,F) (S, ) is measurable, if and only iff:( ,F) (S , ) is itself Letf: R+.

7 Show that the following are equivalent:(i)f:( ,F) (R+,B(R+)) is 4: Measurability12(ii)f:( ,F) (R,B(R)) is measurable(iii)f:( ,F) ( R,B( R)) is measurableExercise ( ,F), (S, ), (S1, 1) be three measurable :( ,F) (S, ) andg:(S, ) (S1, 1) be two For allB S1, show that (g f) 1(B)=f 1(g 1(B))2. Show thatg f:( ,F) (S1, 1) is ( ,F)and(S, ) be two measurable spaces. Letf: Sbe a map. We define: ={B ,f 1(B) F}1. Show thatf 1(S)= . 4: Measurability132. Show that for allB S,f 1(Bc)=(f 1(B)) Show that ifBn S, n 1, thenf 1( + n=1Bn)= + n=1f 1(Bn)4. Show that is a -algebra Prove the following 14 Let( ,F)and(S, )be two measurable spaces, andAbe a set of subsets ofSgenerating , such that = (A).

8 Thenf:( ,F) (S, )is measurable, if and only if: B A,f 1(B) 4: Measurability14 Exercise ( ,T)and(S,TS) be two topological spaces. Letf: Sbe a map. Show that iff:( ,T) (S,TS) is continuous,thenf:( ,B( )) (S,B(S)) is define the following subsets of the power setP( R):C1 ={[ ,c],c R}C2 ={[ ,c[,c R}C3 ={[c,+ ],c R}C4 ={]c,+ ],c R}1. Show thatC2andC4are subsets ofT Show that the elements ofC1andC3are closed in Show that for alli=1,2,3,4, (Ci) B( R).4. LetUbe open in R. Explain whyU Ris open 4: Measurability155. Show that any open subset ofRis a countable union of openbounded intervals Leta<b,a, b R. Show that we have:]a, b[=+ n=1]a, b 1/n]=+ n=1[a+1/n, b[7. Show that for alli=1,2,3,4, ]a, b[ (Ci).

9 8. Show that for alli=1,2,3,4,{{ },{+ }} (Ci).9. Show that for alli=1,2,3,4, (Ci)=B( R)10. Prove the following 4: Measurability16 Theorem 15 Let( ,F)be a measurable space, andf: Rbea map. The following are equivalent:(i)f:( ,F) ( R,B( R)) is measurable(ii) B B( R),{f B} F(iii) c R,{f c} F(iv) c R,{f<c} F(v) c R,{c f} F(vi) c R,{c<f} FExercise ( ,F) be a measurable space. Let (fn)n 1be asequence of measurable mapsfn:( ,F) ( R,B( R)). Letgandhbethe maps defined byg( )=infn 1fn( )andh( )=supn 1fn( ),for all .1. Letc R. Show that{c g}= + n=1{c fn}.2. Letc R. Show that{h c}= + n=1{fn c}. 4: Measurability173. Show thatg, h:( ,F) ( R,B( R)) are 36 Let(vn)n 1be a sequence in R. We define:u = lim infn + vn =supn 1 infk nvk and:w = lim supn + vn =infn 1 supk nvk Then,u, w Rare respectively calledlower limitandupper limitof the sequence(vn)n (vn)n 1be a sequence in 1 we defineun=infk nvkandwn=supk the lower limitand upper limit of (vn)n 1, Show thatun un+1 u, for alln 4: Measurability182.

10 Show thatw wn+1 wn, for alln Show thatun uandwn wasn + .4. Show thatun vn wn, for alln Show thatu Show that ifu=wthen (vn)n 1converges to a limitv R,withu=v= Show that ifa, b Rare such thatu<a<b<wthen for alln 1, there existN1,N2 nsuch thatvN1<a<b< Show that ifa, b Rare such thatu<a<b<wthen thereexist two strictly increasing sequences of integers (nk)k 1and(mk)k 1such that for allk 1, we havevnk<a<b< Show that if (vn)n 1converges to somev R,thenu= 4: Measurability19 Theorem 16 Let(vn)n 1be a sequence in R. Then, the followingare equivalent:(i)lim infn + vn= lim supn + vn(ii)limn + vnexists in which case:limn + vn= lim infn + vn= lim supn + vnExercise , g:( ,F) ( R,B( R)) be two measurablemaps, where ( ,F) is a measurable Show that{f<g}= r Q({f<r} {r<g}).