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modern mathematical statistics with applications

Probability And Measure Wiley Series In Probability And Mathematical Statistics 3rd Edition Patrick Billingsley - Solutions

=+agree, and to take the common value P*( A) = P. (A) as the measure. Since(3.1) and (3.2) agree if and only if(3.3)P*(A) +P*(A)=1, the procedure would be to consider the class of A satisfying (3.3) and use P*( A) as the measure,
=+It turns out to be simpler to impose on A the more stringent requirement that(3.4)
=+P*(ANE) + P*(A'nE) = P*(E)hold for every set E; (3.3) is the special case E = f2, because it will turn out that P*(22) == 1.1 A set A is called P *- measurable if (3.4) holds for all E; let be the class of such sets. What will be shown is that A contains o(7)and that the restriction of P* to
=+The set function P* has four properties that will be needed:(i) P*(Ø) = 0;(ii) P* is nonnegative: P*( A) ≥ 0 for every A CO;(iii) P* is monotone: A C B implies P*(A) ≤ P*(B);(iv) P* is countably subadditive: P*(U ,, A,) ≤ E„, P*( A ,, ).
=+The others being obvious, only (iv) needs proof. For a given €, choose o-sets B, such that A ,, C UR Bnk and __ P(B ., ) < P*(A,) + €2-", which is possible by the definition (3.1). Now U ,, A ,, C Un .. . Bnk, so that P*(U„ A„) ≤ EnP(B„;)
=+By definition, A lies in the class .« of P *- measurable sets if it splits each E in 201 in such a way that P* adds for the pieces-that is, if (3.4) holds.Because of finite subadditivity, this is equivalent to(3.5)
=+P*(ANE) + P*(A‘NE) ≤P*(E).Lemma 1. The class & is a field."It also turns out, after the fact, that (3.3) implies that (3.4) holds for all E anyway; see Problem
=+3.2.Compare the proof on p. 9 that a countable union of negligible sets is negligible.
=+3.1. (a) In the proof of Theorem 3.1 the assumed finite additivity of P is used twice and the assumed countable additivity of P is used once. Where?
=+(b) Show by example that a finitely additive probability measure on a field may not be countably subadditive. Show in fact that if a finitely additive probability measure is countably subadditive, then it is necessarily countably additive as well.
=+(c) Suppose Theorem 2.1 were weakened by strengthening its hypothesis to the assumption that § is a o-field. Why would this weakened result not suffice for the proof of Theorem 3.1?
=+3.2. Let P be a probability measure on a field Fo and for every subset A of 22 define P*( A) by (3.1). Denote also by P the extension (Theorem 3.1) of P to F=0(96).(a) Show that(3.9)P*(A)= inf[ P(B): ACB, BE$]and (see (3.2))(3.10)P. ( A) = sup[P(C): CCA, CE ],
=+and show that the infimum and supremum are always achieved.
=+(b) Show that A is P *- measurable if and only if P. (A) = P*(A).
=+(c) The outer and inner measures associated with a probability measure P on a o-field Fare usually defined by (3.9) and (3.10). Show that (3.9) and (3.10) are the same as (3.1) and (3.2) with $ in the role of Fo.
=+3.3. 2.13 2.15 3.21 For the following examples, describe P* as defined by (3.1)and .A= . A(P*) as defined by the requirement (3.4). Sort out the cases in which
=+P* fails to agree with P on 70 and explain why.
=+(a) Let Fg consist of the sets Ø, {1}, {2, 3), and 2 = {1, 2,3), and define proba-bility measures P, and P2 on , by P,(1) = 0 and P2(2,3} = 0. Note that.((Pf ) and .((Pf) differ.
=+(b) Suppose that 22 is countably infinite, let 9, be the field of finite and cofinite sets, and take P( A) to be 0 or 1 as A is finite or cofinite.
=+(c) The same, but suppose that 22 is uncountable.
=+(d) Suppose that ? is uncountable, let , consist of the countable and the cocountable sets, and take P( A) to be 0 or 1 as A is countable or cocountable.
=+(e) The probability in Problem 2.15.
=+f) Let P(A) = I (wo) for A € Fo, and assume (wg) E o($).
=+3.4. Let f be a strictly increasing, strictly concave function on [0, 00) satisfying f(0) =0. For Ac (0, 1], define P*(A) = f(A*(A)). Show that P* is an outer measure in the sense that it satisfies P*(Ø) - 0 and is nonnegative, monotone, and countably subadditive. Show that A lies in .& (defined
=+(3.4)) if and only if A*(A) or A*(Ac) is 0. Show that P* does not arise from the definition (3.1) for any probability measure P on any field Fo-
=+. Let 2 be the unit square [(x, y): 0 < x, y ≤ 1], let 9 be the class of sets of the form [(x, y): x €A, 0 < y ≤ 1], where A € 8, and let P have value A(A) at this set. Show that (2, 9, P) is a probability measure space. Show for A = [(x, y):0
=+. Let P be a finitely additive probability measure on a field $0. For Ach, in analogy with (3.1) define(3.11)Pº(A) = inf _P(A.), where now the infimum extends over all finite sequences of Fo-sets A, satisfying AC U ,, A ,,. (If countable coverings are allowed, everything is differ-ent. It can
=+(a) Show that Pº(Ø) =0 and that Pº is nonnegative, monotone, and finitely subadditive. Using these four properties of Po, prove: Lemma 1º: 4º is a field.Lemma 2º: If A1, A2 ,... is a finite sequence of disjoint .4º-sets, then for each EcQ, po(En (UA) - IP(EnA).(3.12)
=+Lemma 3º: Pº restricted to the field .4º is finitely additive.(b) Show that if Pº is defined by (3.11) (finite coverings), then: Lemma 4º:F. C.W.º. Lemma 5º: Pº( A) = P(A) for A E F.(c) Define P (A) = 1 - Pº(Ac). Prove that if E CA € 50, then(3.13)P.(E)=P(A)-Pº(A-E).
=+3.7. 2.7 3.61 Suppose that H lies outside the field So, and let 9, be the field generated by FU (H), so that F1 consists of the sets (H NA) U (H' B) with A, B E 90. The problem is to show that a finitely additive probability measure P on So has a finitely additive extension to 51. Define Q on , by
=+(3.14)Q((HOA) U (HenB)) = Pº(HOA) +P (H'nB)for A, B E 70.
=+(a) Show that the definition is consistent.
=+(b) Shows that Q agrees with P on Fo.
=+(c) Show that Q is finitely additive on $1. Show that Q(H) = Pº(H).
=+(d) Define Q' by interchanging the roles of Pº and P. on the right in (3.14).
=+Show that Q' is another finitely additive extension of P to 91. The same is true of any convex combination Q" of Q and Q'. Show that Q"(H) can take any value between P (H) and Po(H).
=+3.8. 1 Use Zorn's lemma to prove a theorem of Tarski: A finitely additive probability measure on a field has a finitely additive extension to the field of all subsets of the space.
=+3.9. 1 (a) Let P be a (countably additive) probability measure on a o-field 9.Suppose that He , and let 91 = 0(U (H). By adapting the ideas in Problem 3.7, show that P has a countably additive extension from F to S.
=+(b) It is tempting to go on and use Zorn's lemma to extend P to a completely additive probability measure on the o-field of all subsets of 2. Where does the obvious proof break down?
=+3.10. 2.17 3.21 As shown in the text, a probability measure space (0, 5, P) has a complete extension-that is, there exists a complete probability measure space
=+(2,5], P1) such that Fc F, and P, agrees with P on $ .
=+(a) Suppose that (2, F2, P2) is a second complete extension. Show by an example in a space of two points that P, and P2 need not agree on the o-field Fin F 2.
=+(b) There is, however, a unique minimal complete extension: Let F+ consist of the sets A for which there exist F-sets B and C such that A & B CC and
=+P(C) = 0. Show that + is a o-field. For such a set A define P+ (A) = P(B).Show that the definition is consistent, that P+ is a probability measure on $ +, and that (n), F+, P+) is complete. Show that, if (0, 91, P1) is any complete extension of (0, 9, P), then F+c $, and P, agrees with Pt on F";
=+(2,7", P+ ) is the completion of (, , P).
=+c) Show that A € $+ if and only if P. (A) = P*( A), where P, and P* are defined by (3.9) and (3.10), and that P*( A) = P.(A) = P*(A) in this case. Thus the complete extension constructed in the text is exactly the completion.
=+3.11. (a) Show that a A-system satisfies the conditions(A4) A, BE.z and AnB =Ø imply AUBE./,(A5) A1, A2 ,... E./ and A ,, 1 A imply A €./,(A6) A1, A2 ,.. . E./ and A„, I A imply A €.f.
=+(b) Show that _/ is a A-system if and only if it satisfies (A,), (X2), and (A,).(Sometimes these conditions, with a redundant (A4), are taken as the definition.)
=+3.12. 2.5 3.11 1 (a) Show that if P is a w-system, then the minimal À-system over 9 coincides with a(9).
=+(b) Let 9 be a wr-system and .& a monotone class. Show that PC.2 does not imply 0(3) C.A.
=+(c) Deduce the T-A theorem from the monotone class theorem by showing directly that, if a A-system ./ contains a w-system 92, then / also contains the field generated by 9.
=+3.13. 2.5 f (a) Suppose that $6) is a field and P, and P2 are probability measures on o(50). Show by the monotone class theorem that if P, and P2 agree on Fon, then they agree on o(5g).
=+(b) Let 9% be the smallest field over the w-system P. Show by the inclusion-exclusion formula that probability measures agreeing on 9 must agree also on
=+50. Now deduce Theorem 3.3 from part (a).
=+3.14. 1.5 2.221 Prove the existence of a Lebesgue set of Lebesgue measure 0 that is not a Borel set.
=+3.15. 1.3 3.6 3.141 The outer content of a set A in (0, 1] is c*( A) = inf E ,, 1/ 1.where the infimum extends over finite coverings of A by intervals / ,,. Thus A is
=+trifling in the sense of Problem 1.3 if and only if c*( A) = 0. Define inner content byc. (A) = 1 - c*(Ac). Show thatc. (A) = sup E ,, I/ ,, I, where the supremum extends over finite disjoint unions of intervals / ,, contained in A (of course the analogue for A , fails). Show thatc. ( A) ≤c*(
=+A trifling set is Jordan measurable. Find (Problem 3.14) a Jordan measurable set that is not a Borel set.
=+Show thatc. ( A) SA (A) ≤ A*(A) ≤c*(A). What happens in this string of inequalities if A consists of the rationals in (0, 2] together with the irrationals in(G, 1]?
=+3.16. 1.5 | Deduce directly by countable additivity that the Cantor set has Lebesgue measure 0.
=+3.17. From the fact that A(x @ A) = À(A), deduce that sums and differences of normal numbers may be nonnormal.
=+3.18. Let H be the nonmeasurable set constructed at the end of the section.
=+(a) Show that, if A is a Borel set and ACH, then A( A) = 0-that is, A . (H) =0.
=+(b) Show that, if A*(E) > 0, then E contains a nonmeasurable subset.
=+3.19. The aim of this problem is the construction of a Borel set A in (0, 1) such that 0
=+(a) It is shown in Example 3.1 how to construct a Borel set of positive Lebesgue measure that is nowhere dense. Show that every interval contains such a set.
=+(b) Let (1 ,, ) be an enumeration of the open intervals in (0, 1) with rational endpoints. Construct disjoint, nowhere dense Borel sets A ,, B1, A2, B2 ,... of positive Lebesgue measure such that A ,, UB ,, CI,
=+(c) Let A - U, A ,. A nonempty open G in (0, 1) contains some / ,,. Show that 0
=+3.20. 1 There is no Borel set A in (0, 1) such that al(I) ≤ A(ANI) sbA(I) for every open interval / in (0, 1), where 0 < a sb < 1. In fact prove:
=+(a) If A(An1) ≤ bA(1) for all / and if b < 1, then A( A)=0. Hint: Choose an open G such that A CG c (0, 1) and A(G) < b 'A(A); represent G as a disjoint union of intervals and obtain a contradiction.
=+(b) If ax(I) ≤ A(An I) for all I and if a > 0, then A( A) = 1.
=+3.21. Show that not every subset of the unit interval is a Lebesgue set. Hint: Show that A* is translation-invariant on 2(0,1); then use the first impossibility theorem(p. 45). Or use the second impossibility theorem.
=+4.1. 2.11 The limits superior and inferior of a numerical sequence {x,) can be defined as the supremum and infimum of the set of limit points-that is, the set of limits of convergent subsequences. This is the same thing as defining(4.30)lim supx, = A V xx n=1k-n and(4.31)lim infx, = V Axx.n
=+Compare these relations with (4.4) and (4.5) and prove that limsup. A. = lim sup [ ...liminf ,, A = lim inf IA. .Prove that lim ,, A ,, exists in the sense of (4.6) if and only if lim ,, I4 (@) exists for each w.
=+4.2. 1 (a) Prove that(lim sup An ) 0 (lim sup Bn ) > lim sup ( A, B,),(lim sup An ) U(lim sup B.) = lim sup ( A, B,),(lim inf A„) (lim inf B.) = lim inf (A,B ,, ),(lim inf A„) U(lim inf B ,, ) C lim inf (A ,, UB ,, ).Show by example that the two inclusions can be strict.(b) The numerical
=+(c) Show that lim sup A" = (lim inf A„) .lim inf At = (lim sup An) , lim sup A ,, - lim inf A ,, = lim sup ( A ,, A"+1)= lim sup ( AçNA,+1).
=+(d) Show that A ., - A and B ,, - B together imply that A ., UB ,, - AUB and A ,, OB, - ANB.
=+4.3. Let A ,, be the square [(x, y): |x| ≤ 1, lyl ≤ 1] rotated through the angle 2Tnf.Give geometric descriptions of limsup ,, A ,, and lim inf A ,, in case(a) 0 = §;
=+(b) 0 is rational;(c) 0 is irrational. Hint: The 2mno reduced modulo 2mr are dense in [0, 2m] if 0
=+is irrational.
=+(d) When is there convergence is the sense of (4.6)?
=+4.4. Find a sequence for which all three inequalities in (4.9) are strict.
=+4.5. (a) Show that lim, P(lim inf, A ,, ( 42) = 0. Hint: Show that lim sup, liminf , A ,, 04% is empty.Put A* = lim sup ,, A ,, and A, = liminf, A ...(b) Show that P(A ,, - A*) -> 0 and P(A, - A.) -> 0.(c) Show that A ,, - A (in the sense that A = A* = A, ) implies P(AA A ,, ) - 0.(d) Suppose
=+4.6. In a space of six equally likely points (a die is rolled) find three events that are not independent even though each is independent of the intersection of the other two.
=+. For events A1 ,..., A ., consider the 2" equations P(B, n . .. n B1) =P(B) ... P(B„) with B, = A, or B; = A; for each i. Show that A1 ,..., A ,, are independent if all these equations hold,
=+4.8. For each of the following classes , describe the -partition defined by (4.16).(a) The class of finite and cofinite sets.(b) The class of countable and cocountable sets.(c) A partition (of arbitrary cardinality) of 2.(d) The level sets of sin x (2 = R').
=+e) The o-field in Problem 3.5.4.9. 2.9 2.101 In connection with Example 4.8 and Problem 2.10, prove these facts:
=+(a) Every set in o(s ) is a union of ofequivalence classes.(b) If =[A .: 0 € 0], then the ofequivalence classes have the form n , Be, where for each 0, Bo is Ag Or AS.(c) Every finite o-field is generated by a finite partition of Q.(d) If 96 is a field, then each singleton, even each finite
=+4.10. 3.21 There is in the unit interval a set H that is nonmeasurable in the extreme sense that its inner and outer Lebesgue measures are 0 and 1 (see (3.9)and (3.10)): A,(H)=0 and A*(H)=1. See Problem 12.4 for the construction.Let 2 = (0, 1], let & consist of the Borel sets in 22, and let H be
=+The construction proves this: There exist a probability space (9, 5, P), a o-field & in F, and a set H in F, such that P(H) =}, H and & are independent, and & is generated by a countable subclass and contains all the singletons.
=+Example 4.10 is somewhat similar, but there the o-field & is not countably generated and each set in it has probability either 0 or 1. In the present example 3 is countably generated and P(G) assumes every value between 0 and 1 as G ranges over . Example 4.10 is to some extent unnatural because
=+4.11. (a) If A1, A2 ,... are independent events, then P(("- ¡ A,)= IIm_ P(A,) and P(U",A.)=1-IT ;_; (1-P(A)). Prove these facts and from them derive the second Borel-Cantelli lemma by the well-known relation between infinite· series and products.
=+(b) Show that P(lim sup ,, A.) = 1 if for each k the series E ,., P(A,JAG.. . OAG ,_ 1) diverges. From this deduce the second Borel-Cantelli lemma once again.
=+(c) Show by example that P(lim sup ,, A ,, ) = 1 does not follow from the diver-gence of E ,, P(A,LAGO · · · NA" _, ) alone.
=+(d) Show that P(lim sup ,, A„) = 1 if and only if E ,, P( ANA ,, ) diverges for each A of positive probability.
=+e) If sets A ,, are independent and P(A„) < 1 for all n, then P[ A ,, i.o.]=1 if and only if P(U ,, A,) =1.
=+4.12. (a) Show (see Example 4.21) that log2 n + log2 log 2 n + 0 log, log, log2 n is an outer boundary if 0 > 1. Generalize.
=+(b) Show that log2 n + log2 log, log, n is an inner boundary.
=+4.13. Let o be a positive function of integers, and define B, as the set of x in (0, 1)such that [x -p/2'\< 1/2'(2') holds for infinitely many pairs p, i. Adapting the proof of Theorem 1.6, show directly (without reference to Example 4.12)that E;1/4(24)
=+4.14. 2.191 Suppose that there are in (2, 5, P) independent events A1, A2 ,...such that, ifa, = min( P(A ), 1 - P(A)), then Ea ,, = ", Show that P is nonatomic.

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