Note 9. To show the existence of a sample space and probability measure with such
27. For any real or complex number z = 1 and any positive integer N, derive the geomet-
ric series formula
N−1
∑
k=0 zk = 1− z N 1− z , z = 1.Hint: Let SN:= 1 + z + ··· + zN−1, and show that SN− zSN = 1 − zN. Then solve for
SN. Remark. If |z| < 1, |z|N→ 0 as N → ∞. Hence, ∞
∑
k=0 zk = 1 1− z, for |z| < 1.28. Let Ω := {1,...,6}. If p(ω) = 2 p(ω− 1) forω= 2,...,6, and if ∑6
ω=1p(ω) = 1,
1.4: Axioms and properties of probability
29. Let A and B be events for which P(A), P(B), and P(A ∪ B) are known. Express the following in terms of these probabilities:
(a) P(A ∩ B). (b) P(A ∩ Bc). (c) P(B ∪ (A ∩ Bc)). (d) P(Ac∩ Bc).
30. Let Ω be a sample space equipped with two probability measures, P1andP2. Given
any 0≤λ≤ 1, show that if P(A) :=λP1(A)+(1−λ)P2(A), then P satisfies the four
axioms of a probability measure.
31. Let Ω be a sample space, and fix any pointω0∈ Ω. For any event A, put
µ(A) :=
1, ω0∈ A, 0, otherwise. Show thatµsatisfies the axioms of a probability measure.
32. Suppose that instead of axiom (iii) of Section 1.4, we assume only that for any two disjoint events A and B,P(A∪B) = P(A)+P(B). Use this assumption and inductionj on N to show that for any finite sequence of pairwise disjoint events A1,...,AN,
P N n=1 An =
∑
N n=1 P(An).Using this result for finite N, it is not possible to derive axiom (iii), which is the assumption needed to derive the limit results of Section 1.4.
33. The purpose of this problem is to show that any countable union can be written as a
union of pairwise disjoint sets. Given any sequence of sets Fn, define a new sequence
by A1:= F1, and
An := Fn∩ Fnc−1∩ ··· ∩ F1c, n ≥ 2.
Note that the Anare pairwise disjoint. For finite N≥ 1, show that N n=1 Fn = N n=1 An.
Also show that
∞ n=1 Fn = ∞ n=1 An.
jIn this case, using induction on N means that you first verify the desired result for N= 2. Second, you assume
34. Use the preceding problem to show that for any sequence of events F n, P ∞ n=1 Fn = lim N→∞P N n=1 Fn .
35. Use the preceding problem to show that for any sequence of events G
n, P ∞ n=1 Gn = lim N→∞P N n=1 Gn .
36. The finite union bound.Show that for any finite sequence of events F1,...,FN,
P N n=1 Fn ≤
∑
N n=1 P(Fn).Hint: Use the inclusion–exclusion formula (1.12) and induction on N. See the last
footnote for information on induction.
37. The infinite union bound.Show that for any infinite sequence of events F
n, P ∞ n=1 Fn ≤
∑
∞ n=1 P(Fn).Hint: Combine Problems 34 and 36. 38. First Borel–Cantelli lemma.Show that if B
nis a sequence of events for which
∞
∑
n=1P(B n) < ∞, (1.36) then P ∞ n=1 ∞ k=n Bk = 0.Hint: Let G :=∞n=1Gn, where Gn:= ∞
k=nBk. Now use Problem 35, the union bound
of the preceding problem, and the fact that (1.36) implies lim N→∞ ∞
∑
n=N P(Bn) = 0.39. Let Ω = [0,1], and for A ⊂ Ω, put P(A) :=
A1 dω. In particular, this impliesP([a,b])
= b−a. Consider the following sequence of sets. Put A0:= Ω = [0,1]. Define A1⊂ A0
by removing the middle third from A0. In other words, A1 = [0,1/3] ∪ [2/3,1].
Now define A2⊂ A1by removing the middle third of each of the intervals making up A1. An easy way to do this is to first rewrite
Then
A2 =
[0,1/9] ∪ [2/9,3/9]∪[6/9,7/9] ∪ [8/9,9/9].
Similarly, define A3by removing the middle third from each of the above four inter-
vals. Thus,
A3 := [0,1/27] ∪ [2/27,3/27]
∪ [6/27,7/27] ∪ [8/27,9/27] ∪ [18/27,19/27] ∪ [20/27,21/27] ∪ [24/27,25/27] ∪ [26/27,27/27]. Continuing in this way, we can define A4, A5, . . . .
(a) ComputeP(A0), P(A1), P(A2), and P(A3).
(b) What is the general formula forP(An)?
(c) The Cantor set is defined by A :=∞
n=0An. FindP(A). Justify your answer.
40. This problem assumes you have read Note 1. Let A
1,...,Anbe a partition ofΩ. If
C := {A1,...,An}, show thatσ(C ) consists of the empty set along with all unions of
the form
i
Aki
where kiis a finite subsequence of distinct elements from{1,...,n}.
41. This problem assumes you have read Note 1. Let Ω := [0,1), and for n = 1,2,...,
letCndenote the partition
Cn := k− 1 2n , k 2n ,k = 1,...,2n.
LetAn:=σ(Cn), and put A :=∞n=1An. Determine whether or notA is aσ-field.
42. This problem assumes you have read Note 1. Let Ω be a sample space, and let X:Ω → IR, where IR denotes the set of real numbers. Suppose the mapping X takes
finitely many distinct values x1,...,xn. Find the smallestσ-fieldA of subsets of Ω
such that for all B⊂ IR, X−1(B) ∈ A . Hint: Problems 14 and 15.
43. This problem assumes you have read Note 1. Let Ω := {1,2,3,4,5}, and put A :=
{1,2,3} and B := {3,4,5}. Put P(A) := 5/8 and P(B) := 7/8.
(a) FindF :=σ({A,B}), the smallestσ-field containing the sets A and B. (b) ComputeP(F) for all F ∈ F .
(c) Trick question. What isP({1})?
44. This problem assumes you have read Note 1. Show that aσ-field cannot be count-
ably infinite; i.e., show that if aσ-field contains an infinite number of sets, then it contains an uncountable number of sets.
45. This problem assumes you have read Note 1.
(a) LetAαbe any indexed collection ofσ-fields. Show thatαAαis also aσ-field. (b) Illustrate part (a) as follows. LetΩ := {1,2,3,4},
A1:=σ({1},{2},{3,4}) and A2:=σ({1},{3},{2,4}).
FindA1∩ A2.
(c) LetC be any collection of subsets of Ω, and letσ(C ) denote the smallestσ- field containingC . Show that
σ(C ) =
A :C ⊂A
A ,
where the intersection is over allσ-fieldsA that contain C .
46. This problem assumes you have read Note 1. Let Ω be a nonempty set, and let F
andG beσ-fields. IsF ∪ G aσ-field? If “yes,” prove it. If “no,” give a counterex- ample.
47. This problem assumes you have read Note 1. Let Ω denote the positive integers.
LetA denote the collection of all subsets A such that either A is finite or Acis finite.
(a) Let E denote the positive integers that are even. Does E belong toA ?
(b) Show thatA is closed under finite unions. In other words, if A1,...,Anare in
A , show thatn
i=1Aiis also inA .
(c) Determine whether or notA is aσ-field.
48. This problem assumes you have read Note 1. Let Ω be an uncountable set. Let A
denote the collection of all subsets A such that either A is countable or Acis countable. Determine whether or notA is aσ-field.
49. The Borelσσ-field.This problem assumes you have read Note 1. LetB denote the
smallestσ-field containing all the open subsets of IR := (−∞,∞). This collection B is called the Borelσσ-field. The sets inB are called Borel sets. Hence, every open
set, and every open interval, is a Borel set.
(a) Show that every interval of the form(a,b] is also a Borel set. Hint: Write (a,b] as a countable intersection of open intervals and use the properties of aσ-field. (b) Show that every singleton set{a} is a Borel set.
(c) Let a1,a2,... be distinct real numbers. Put A :=
∞
k=1
{ak}.
(d) Lebesgue measureλ on the Borel subsets of (0,1) is a probability measure that is completely characterized by the property that the Lebesgue measure of an open interval(a,b) ⊂ (0,1) is its length; i.e.,λ(a,b)= b − a. Show that
λ(a,b]is also equal to b− a. Findλ({a}) for any singleton set. If the set A in part (c) is a Borel set, computeλ(A).
Remark. Note 5 in Chapter 5 contains more details on the construction of prob-
ability measures on the Borel subsets of IR.
50. The Borelσσ-field, continued.This problem assumes you have read Note 1.
Background: Recall that a set U⊂ IR is open if for every x ∈ U, there is a positive
numberεx, depending on x, such that(x −εx,x +εx) ⊂ U. Hence, an open set U can
always be written in the form
U =
x∈U
(x −εx,x +εx).
Now observe that if(x −εx,x +εx) ⊂ U, we can find a rational number qxclose to x
and a rational numberρx<εxsuch that
x∈ (qx−ρx,qx+ρx) ⊂ (x −εx,x +εx) ⊂ U.
Thus, every open set can be written in the form
U =
x∈U
(qx−ρx,qx+ρx),
where each qxand eachρxis a rational number. Since the rational numbers form a
countable set, there are only countably many such intervals with rational centers and rational lengths; hence, the union is really a countable one.
Problem: Show that the smallestσ-field containing all the open intervals is equal to the Borelσ-field defined in Problem 49.
1.5: Conditional probability