Problems and Solutions in - Mathematics - University of Hawaii

1.3 1998 April 3 1 REAL ANALYSIS
1.3 1998 April 3
Instructions Do at least four problems in Part A, and at least two problems in Part B.
PART A
1. Let {xn} ∞ n=1 be a bounded sequence of real numbers, and for each positive n define
(a) Explain why the limit ℓ = limn→∞ ˆxn exists.
ˆxn = sup xk
k≥n
(b) Prove that, for any ɛ > 0 and positive integer N, there exists an integer k such that k ≥ N and |xk − ℓ| < ɛ.
2. Let C be a collection of subsets of the real line R, and define
Aσ(C) = {A : C ⊂ A and A is a σ-algebra of subsets of R}.
(a) Prove that Aσ(C) is a σ-algebra, that C ⊂ Aσ(C), and that Aσ(C) ⊂ A for any other σ-algebra A containing
all the sets of C.
(b) Let O be the collection of all finite open intervals in R, and F the collection of all finite closed intervals in R.
Show that
Aσ(O) = Aσ(F ).
3. Let (X, A, µ) be a measure space, and suppose X = ∪nXn, where {Xn} ∞ n=1 is a pairwise disjoint collection of
measurable subsets of X. Use the monotone convergence theorem and linearity of the integral to prove that, if f is
a non-negative measurable real-valued function on X,
f dµ =
f dµ.
X
n
Solution: 15 Define fn = n k=1 fχXk = fχ∪n 1
Xk . Then it is clear that the hypotheses of the monotone convergence
theorem are satisfied. That is, for all x ∈ X,
(i) 0 ≤ f1(x) ≤ f2(x) ≤ · · · ≤ f(x), and
(ii) limn→∞ fn(x) = f(x)χX(x) = f(x).
15Note that the hypotheses imply ν(E) = E f dµ is a measure (problem 4, Nov. ’91), from which the desired conclusion immediately follows.
Of course, this does not answer the question as stated, since the examiners specifically require the use of the MCT and linearity of the integral.
12
Xn