Lecture 7: Borel Sets and Lebesgue Measure

Transcription

1 EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture, we discuss the case where the sample space is ucoutable. This case is more ivolved tha the case of a coutable sample space, maily because it is ofte ot possible to assig probabilities to all subsets of Ω. Istead, we are forced to work with a smaller σ-algebra. We cosider assigig a uiform probability measure o the uit iterval. 7. Ucoutable sample spaces Cosider the experimet of pickig a real umber at radom from Ω = [0, ], such that every umber is equally likely to be picked. It is quite apparet that a simple strategy of assigig probabilities to sigleto subsets of the sample space gets ito difficulties quite quickly. Ideed, i) If we assig some positive probability to each elemetary outcome, the the probability of a evet with ifiitely may elemets, such as A = {, 2, 3, }, would become ubouded. ii) If we assig zero probability to each elemetary outcome, this aloe would ot be sufficiet to determie the probability of a ucoutable subset of Ω, such as [ 2, 3] 2. This is because probability measures are ot additive over ucoutable disjoit uios of sigletos i this case). Thus, we eed a differet approach to assig probabilities whe the sample space is ucoutable, such as Ω = [0, ]. I particular, we eed to assig probabilities directly to specific subsets of Ω. Ituitively, we would like our uiform measure µ o [0, ] to possess the followig two properties. i) µ a, b)) = µ a, b]) = µ [a, b)) = µ [a, b]) ii) Traslatioal Ivariace. That is, if A [0, ], the for ay x Ω, µ A x) = µ A) where, the set A x is defied as A x = {a + x a A, a + x } {a + x a A, a + x > } However, the followig impossibility result asserts that there is o way to cosistetly defie a uiform measure o all subsets of [0, ]. Theorem 7. Impossibility Result) There does ot exist a defiitio of a measure µ A) for all subsets of [0, ] satisfyig i) ad ii). Proof: Refer propositio.2.6 i []. Therefore, we must compromise, ad cosider a smaller σ-algebra that cotais certai ice subsets of the sample space [0, ]. These ice subsets are the itervals, ad the resultig σ-algebra is called the Borel σ-algebra. Before defiig Borel sets, we itroduce the cocept of geeratig σ-algebras from a give collectio of subsets. 7-

2 7-2 Lecture 7: Borel Sets ad Lebesgue Measure 7.2 Geerated σ-algebra ad Borel sets The σ-algebra geerated by a collectio of subsets of the sample space is the smallest σ-algebra that cotais the collectio. More formally, we have the followig theorem. Theorem 7.2 Let C be a arbitrary collectio of subsets of Ω, the there exists a smallest σ-algebra, deoted by σ C), that cotais all elemets of C. That is, if H is ay σ-algebra such that C H, the σ C) H. σ C) is called the σ-algebra geerated by C. Proof: Let {F i, i I} deote the collectio of all σ-algebras that cotai C. Clearly, the collectio {F i, i I} is o-empty, sice it cotais at least the power set, 2 Ω. Cosider the itersectio F i. Sice the itersectio of σ-algebras results i a σ-algebra homework problem!) ad the itersectio cotais C, it follows that F i is a σ-algebra that cotais C. Fially, if C H, the H is oe of F i s for some i I. i I Hece F i is the smallest σ-algebra geerated by C. i I Ituitively, we ca thik of C as beig the collectio of subsets of Ω which are of iterest to us. The, σc) is the smallest σ-algebra cotaiig all the iterestig subsets. We are ow ready to defie Borel sets. Defiitio 7.3 a) Cosider Ω = 0, ]. Let C 0 be the collectio of all ope itervals i 0, ]. The σ C 0 ), the σ - algebra geerated by C 0, is called the Borel σ - algebra. It is deoted by B 0, ]). b) A elemet of B 0, ]) is called a Borel-measurable set, or simply a Borel set. i I Thus, every ope iterval i 0, ] is a Borel set. We ext prove that every sigleto set i 0, ] is a Borel set. Lemma 7.4 Every sigleto set {b}, 0 < b, is a Borel set, i.e., {b} B 0, ]). Proof: Cosider the collectio of sets set { b, b + } ),. By the defiitio of Borel sets, b, b + ) B 0, ]). Usig the properties of σ-algebra, = = = b, b + ) c B 0, ]) b, b + ) c B 0, ]) b, b + ) ) c B 0, ]) b, b + ) B 0, ]). 7.)

3 Lecture 7: Borel Sets ad Lebesgue Measure 7-3 Next, we claim that {b} = b, b + ). 7.2) i.e., b is the oly elemet i b, b + ). We prove this by cotradictio. Let h be a elemet i b, b + ) other tha b. For every such h, there exists a large eough 0 such that h / b 0, b + ). 0 This implies h / b, b + ). Usig 7.) ad 7.2), thus, proves that {b} B 0, ]). As a immediate cosequece to this lemma, we see that every half ope iterval, a, b], is a Borel set. This follows from the fact that a, b] = a, b) {b}, ad the fact that a coutable uio of Borel sets is a Borel set. For the same reaso, every closed iterval, [a, b], is a Borel set. Note: Arbitrary uio of ope sets is always a ope set, but ifiite itersectios of ope sets eed ot be ope. Further readig for the ethusiastic: try Wikipedia for a start) No-Borel sets No-measurable sets Vitali set) Baach-Tarski paradox a bizzare pheomeo about cuttig up the surface of a sphere. See https: // The cardiality of the Borel σ-algebra o the uit iterval) is the same as the cardiality of the reals. Thus, the Borel σ-algebra is a much smaller collectio tha the power set 2 [0,]. See https: //math.dartmouth.edu/archive/m03f08/public_html/borel-sets-sol.pdf 7.3 Caratheodory s Extesio Theorem I this sectio, we discuss a formal procedure to defie a probability measure o a geeral measurable space Ω, F). Specifyig the probability measure for all the elemets of F directly is difficult, so we start with a smaller collectio F 0 of iterestig subsets of Ω, which eed ot be a σ-algebra. We should take F 0 to be rich eough, so that the σ-algebra it geerates is same as F. The we defie a fuctio P 0 : F 0 [0, ], such that it correspods to the probabilities we would like to assig to the iterestig subsets i F 0. Uder certai coditios, this fuctio P 0 ca be exteded to a legitimate probability measure o Ω, F) by usig the followig fudametal theorem from measure theory. Theorem 7.5 Caratheodory s extesio theorem) Let F 0 be a algebra of subsets of Ω, ad let F = σ F 0 ) be the σ-algebra that it geerates. Suppose that P 0 is a mappig from F 0 to [0, ] that satisfies P 0 Ω) =, as well as coutable additivity o F 0. The, P 0 ca be exteded uiquely to a probability measure o Ω, F). That is, there exists a uique probability measure P o Ω, F) such that P A) = P 0 A) for all A F 0.

4 7-4 Lecture 7: Borel Sets ad Lebesgue Measure Proof: Refer Appedix A of [2]. We use this theorem to defie a uiform measure o 0, ], which is also called the Lebesgue measure. 7.4 The Lebesgue measure Cosider Ω = 0, ]. Let F 0 cosist of the empty set ad all sets that are fiite uios of the itervals of the form a, b]. A typical elemet of this set is of the form F = a, b ] a 2, b 2 ]... a, b ] where, 0 a < b a 2 < b 2... a < b ad N. Lemma 7.6 a) F 0 is a algebra b) F 0 is ot a σ-algebra c) σ F 0 ) = B Proof: a) By defiitio, Φ F 0. Also, Φ C = 0, ] F 0. The complemet of a, b ] a 2, b 2 ] is 0, a ] b, a 2 ] b 2, ], which also belogs to F 0. Furthermore, the uio of fiitely may sets each of which are fiite uios of the itervals of the form a, b], is also a set which is the uio of fiite umber of itervals, ad thus belogs to F 0. b) To see this, ote that ] 0, + F 0 for every, but ] 0, + = 0, ) / F 0. c) First, the ull set is clearly a Borel set. Next, we have already see that every iterval of the form a, b] is a Borel set. Hece, every elemet of F 0 other tha the ull set), which is a fiite uio of such itervals, is also a Borel set. Therefore, F 0 B. This implies σ F 0 ) B. Next we show that B σ F 0 ). For ay iterval of the form a, b) i C 0, we ca write a, b) = ] ) a, b Ω. Sice every iterval of the form a, b ] F0, a coutable umber of uios of such itervals belogs to σ F 0 ). Therefore, a, b) σ F 0 ) ad cosequetly, C 0 σ F 0 ). This gives σ C 0 ) σ F 0 ). Usig the fact that σ C 0 ) = B proves the required result. For every F F 0 of the form F = a, b ] a 2, b 2 ]... a, b ], we defie a fuctio P 0 : F 0 [0, ] such that P 0 Φ) = 0 ad P 0 F ) = b i a i ).

5 Lecture 7: Borel Sets ad Lebesgue Measure 7-5 Note that P 0 Ω) = P 0 0, ]) =. Also, if a, b ], a 2, b 2 ],..., a, b ] are disjoit sets, the ) P 0 a i, b i ]) = P 0 a i, b i ]) = b i a i ) implyig fiite additivity of P 0. It turs out that P 0 is coutably additive o F 0 as well i.e., if a, b ], a 2, b 2 ],... are disjoit sets such that ) a i, b i ]) F 0, the P 0 a i, b i ]) = P 0 a i, b i ]) = b i a i ). The proof is o-trivial ad beyod the scope of this course see [Williams] for a proof). Thus, i view of Theorem 7.5, there exists a uique probability measure P o 0, ], B) which is the same as P 0 o F 0. This uique probability measure o 0, ] is called the Lebesgue or uiform measure. The Lebesgue measure formalizes the otio of legth. This suggests that the Lebesgue measure of a sigleto should be zero. This ca be show as follows. Let b 0, ]. Usig 7.2), we write P {b}) = P b ] ), b Ω Let A = b, b]. For each, the lebesgue measure of A is P A ) = 7.3) Sice A is a decreasig sequece of ested sets, ) P {b}) =P A = lim P A ) = lim where the secod equality follows from the cotiuity of probability measures. =0 Sice ay coutable set is a coutable uio of sigletos, the probability of a coutable set is zero. For example, uder the uiform measure o 0, ], the probability of the set of ratioals is zero, sice the ratioal umbers i 0, ] form a coutable set. For Ω = 0, ], the Lebesgue measure is also a probability measure. For other itervals for example Ω = 0, 2]), it will oly be a fiite measure, which ca be ormalized as appropriate to obtai a uiform probability measure. Defiitio 7.7 Let Ω, F, P) be a probability space. PA) =. A evet A is said to occur almost surely a.s) if Cautio: PA) = does ot mea A = Ω.

6 7-6 Lecture 7: Borel Sets ad Lebesgue Measure Lebesgue Measure of the Cator set: Cosider the cator set K. It is created by repeatedly removig the ope middle thirds of a set of lie segmets. Cosider its complemet. It cotais coutable umber of disjoit itervals. Hece we have: PK c ) = = =. Therefore PK) = 0. It is very iterestig to ote that though the Cator set is equicardial with 0, ], its Lebesgue measure is equal to 0 while the Lebesgue measure of 0, ] is equal to. We ow exted the defiitio of Lebesgue measure o [0, ] to the real lie, R. We first look at the defiitio of a Borel set o R. This ca be doe i several ways, as show below. Defiitio 7.8 Borel sets o R: Let C be a collectio of ope itervals i R. The BR) = σc) is the Borel set o R. Let D be a collectio of semi-ifiite itervals {, x]; x R}, the σd) = BR). A R is said to be a Borel set o R, if A, + ] is a Borel set o, + ] Z. Exercise: Verify that the three statemets are equivalet defiitios of Borel sets o R. Defiitio 7.9 Lebesgue measure of A R: λa) = P A, + ]) = Theorem 7.0 R, BR), λ) is a ifiite measure space. Proof: We eed to prove followig: λr) = λφ) = 0 The coutable additivity property We see that P R, + ]) =, I Hece we have λr) = = = Now cosider Φ, + ]. This is a ull set for all. Hece we have, P Φ, + ]) = 0, I which implies, λφ) = P Φ, + ]) = 0 =

7 Lecture 7: Borel Sets ad Lebesgue Measure 7-7 We ow eed to prove the coutable additivity property. For this we cosider A i BR) such that the sequece A, A 2,..., A,... are arbitrary pairwise disjoit sets i BR). Therefore we obtai, λ A i ) = = = = P = = A i, + ]) P A i, + ]) P A i, + ]) The secod equality above comes from the fact that the probability measure has coutable additivity property. The last equality above comes from the fact that the summatios ca be iterchaged from Fubii s theorem). We also have the followig: λa i ) = = P A i, + ]) We ow immediately see that Hece proved. λ A i ) = λa i ) 7.5 Exercises. Let F be a σ-algebra correspodig to a sample space Ω. Let H be a subset of Ω that does ot belog to F. Cosider the collectio G of all sets of the form H A) H c B), where A ad B F. a) Show that H A G. b) Show that G is a σ-algebra. 2. Show that C = σc) iff C is a σ-algebra. 3. Let C ad D be two collectios of subsets of Ω such that C D. Prove that σc) σd). 4. Prove that the followig subsets of 0, ] are Borel-measurable. a) ay coutable set b) the set of irratioal umbers c) the Cator set Hit: rather tha defiig it i terms of terary expasios, it s easier to use the equivalet defiitio of the Cator set that ivolves sequetially removig the middle-third ope itervals; see Wikipedia for example). d) The set of umbers i 0, ] whose decimal expasio does ot cotai Let B deote the Borel σ-algebra as defied i class. Let C c deote the set of all closed itervals cotaied i 0, ]. Show that σc c ) = B. I other words, we could have very well defied the Borel σ-algebra as beig geerated by closed itervals, rather tha ope itervals.

8 7-8 Lecture 7: Borel Sets ad Lebesgue Measure 6. Let Ω = [0, ], ad let F 3 cosist of all coutable subsets of Ω, ad all subsets of Ω havig a coutable complemet. It ca be show that F 3 is a σ-algebra Refer Lecture 4, Exercises, 6d)). Let us defie PA) = 0 if A coutable, ad PA) = if A has a coutable complemet. Is Ω, F 3, P) a legitimate probability space? 7. We have see i 4c) that the Cator set is Borel-measurable. Show that the Cator set has zero Lebesgue measure. Thus, although the Cator set ca be put ito a bijectio with [0, ], it has zero Lebesgue measure! Refereces [] Rosethal, J. S. 2006). A first look at rigorous probability theory Vol. 2). Sigapore: World Scietific. [2] Williams, D. 99). Probability with martigales. Cambridge uiversity press.