1HomeWork 2 DUE DATE : OCT 19, 2010MATH 660 Instructor: Dr. RON SAHOODirection: This homework worths 100 points. To receive full credit, answer each problem correctlyand must show all works.1. Let X = {1, 2, 3}, C1= {{1}, {2, 3}, ∅, X}, and C2= {{3}, {1, 2}, ∅, X}. Verify that C1and C2are both σ-algebras but C1∪ C2is not a σ-algebra.2. Let X and Y be non empty sets, and (Y, F) be a measurable space. For any functionf : X → Y , show that the collection {f−1(A) | ∀A ∈ F} is a σ-algebra in X.3. Let (X, C) be a measurable space. If the σ-algebra C has a finite number of subsets of X in it,then C is also a topology on X.4. Let X = {a, b, c, d}, C = {X, ∅, {a}, {b, c, d}} and let F = P(X), the set of all subsets of X.Define the functions f, g : X → X byf(x) = a for x ∈ Xandg(x) =(a if x = a, bc if x = c, d.Show that f is (C, F)-measurable but g is not (C, F)-measurable.5. (a) Let f : X →¯R be a constant function, f(x) = α for all x ∈ X. Show that f is measurable.(b) Let f : X →¯R be a measurable function and α ∈ R be any real constant. Show that αfis also measurable.6. Let (X, C) be a measurable space and let f, g : X → R be any two measurable real-valuedfunctions. Show that the functions f + g, f − g and fg are measurable.7. Define a relation on [0, 1] as follows: for x, y ∈ [0, 1] we say x is related to y, written as x ∼ y,if x − y ∈ Q. Prove the followings:2(a) Show that ∼ is an equivalence relation on [0, 1].(b) Let {Eα}α∈Idenote the set of equivalence classes of elements of [0, 1]. Using the axiomof choice, choose exactly one element xα∈ Eαfor every α ∈ I and construct the set E ={xα| α ∈ I}. Let r1, r2, r3, ..., rn, .... denote an enumeration of the rationals in [−1, 1]. LetEn:= rn+ E, n = 1, 2, 3, ...Show that En∩ Em= ∅ for n 6= m and En⊆ [−1, 2] for every n. Deduce[0, 1] ⊆∞[n=1En⊆ [−1, 2].(c) Show that E is not Lebesgue measurable.8. Let X be a set and A, B, C ⊂ X. The function χA: X → {0, 1} defined byχA(x) =(1 if x ∈ A0 if x 6∈ Ais called the characteristic (or indicator) function of A. Show(a) χA∩B= χA· χB, where χA· χB(x) = χA(x) χB(x).(b) χA∪B= χA+ χB− χA∩B.9. Let (X, C, λ) be a measure space, ψ ∈ M+(X, C) be a simple function and c ≥ 0 be a realnumber. Then show thatZc ψ dλ = cZψ dλ.10. Let (X, C, λ) be a measure space, ψ, ξ ∈ M+(X, C) be simple functions. Then show thatZ(ψ + ξ) dλ =Zξ dλ +Zψ dλ.11. Let (X, C, λ) be a measure space, and let ψ ∈ M+(X, C) be a simple function, and A ∈ C.DefineRAψ dλ :=Rψ χAdλ, where χAis the characteristic function of the set A. Show that theset function µ(A) defined byµ(A) =ZAψ dλis a measure on the σ-algebra C.312. Let (fn) be a sequence of elements in M+(X, C). By lim inf fnwe mean the function Kdefined byK(x) = lim inf fn(x).Prove the followings:(1). [Fatou’s Lemma] Let (fn) be a sequence of elements in M+(X, C) and let λ be a measure onC. ThenZ(lim inf fn) dλ ≤ lim infZfndλ.(2). Let f ∈ M+(X, C) and let λ be a measure on C. If E = {x ∈ X | f (x) > 0} is an element ofC and λ(E) = 0, thenRf dλ = 0.Hint: Define fn(x) = n χE(x). Show that this sequence satisfies the hypothesis of Fatou’s Lemmaand that f ≤ lim inf fn. Use Fatou’s Lemma and Theorem 2.2.20 to complete the proof.13. Let (X, C, λ) be a measure space. Prove that the indicator function χAis non-measurable,where A is a non-measurable set of X.14. Let (X, C, λ) be a measure space and let fn: X → R, n = 1, 2, 3, .... be a monotonic sequenceof measurable functions such that lim fn(x) = f(x). Prove that f (x) is measurable.15. Let f(x) be 1, if x is a rational number, 0 otherwise. Prove that f(x) is not Riemannintegrable on the interval [0, 1].September 28,