MATH 741 Measure and Probability Homework 1. Homework 1, due Monday, January 16th a. Suppose f : X → Y. For A , ∈ ⊂ 2 Y , prove that f −1 ∩ ∈ A ∩ ∈ f −1 A . ∩ ∈ f A ? (In this class, to answer the latter Is it true in general that f ∩ ∈ A question you should either prove the statement is true in general, or provide a specific counterexample to show that it isn’t.) b. Consider a non-empty set X and a field ℱ ⊂ 2 X . Prove that ℱ is a ring in the algebraic structures sense of the word, with ring addition of two members A and B of ℱ being AΔB, the set symmetric difference, and ring multiplication being A ∩ B, the set intersection. What are the identity and inverse elements of this ring? c. Suppose that ~ is an equivalence relation on a set X. Prove that the equivalence classes under ~ form a partition of X. d. Suppose O ⊂ is open. Prove that there exists a unique, mutually disjoint, countable collection of open intervals I n a n , b n , n ∈ ℕ, a n ∈ ̄ , b n ∈ ̄ such that O ⊔ n 1 I n . Hints: How do you define open interval? For each x ∈ O, is there a maximal open interval I x ⊂ O? If so, how many of these can be distinct? e. Suppose that X, d is a metric space, and x n n≥1 ⊂ X is a sequence in X. Prove that if lim n x n exists, then it is unique. f. Suppose that X, d and Y, are metric spaces, and that f : X, d → Y, is continuous. Prove that the , definition of continuity of f and the open sets definition of continuity of f are equivalent, i.e. each implies the other. g. Can you find a set X and a sequence of sets A n , n ∈ ℕ ⊂ 2 X such that lim inf A n ∅ lim sup A n X? 2. Homework 2, due Monday, January 23rd a. Use the definition of m ∗ to prove directly that m ∗ ℂ 3 0. Note that as of this 1 − m ∗ 0, 1 − ℂ 3 , so that homework assignment we don’t know that m ∗ ℂ 3 method of proof is not available to use here. 3. Homework 3, due Monday, January 30th l a, b . a. Suppose that − a b . Prove that m ∗ a, b ∗ b. Suppose that − a b . Prove that m a, b l a, b . ∗ c. Suppose that − a b . Prove that m a, b l a, b . ∗ d. Suppose that E ⊂ with m E 0. Prove that E is measurable. 4. Homework 4, due Monday, February 6th a. Let X be a countably infinite set and C A⊂X:#A . Describe C . b. Let Y be an uncountably infinite set and D A⊂Y:#A . Describe D . c. Let Z be an uncountably infinite set and F A⊂Z:#A or # A ℵ0 . Describe F . d. Prove that a, b ⊂ is both an F set and a G set. e. Prove that ℂ 3 is both an F set and a G set. f. There are sets that are not both F and G sets. is one such set, but proving this uses point set topology beyond the scope of this course. Find a source online that proves this and site the source as your answer to this question. (I want you to believe there are such sets but I don’t want you to spend hours proving it, hence this slightly unusual homework question.) 5. Homework 5, due Monday, February 13th 0 a. → Suppose F : if x 0 1, 2 3 3 x 2 if 0 ≤ x ≤ 1 . Find m F , Fx 1 if 1 , m F 0 , and x mF ℂ3 . → 0 if x 0 1, 2 3 3 b. Suppose G : c. mG ℂ3 . Suppose F ℂ : 0, 1 → 0, 1 is the middle thirds Cantor function (aka Devil’s Staircase , Gx 1 if 0 ≤ x . Find m G , m G 0 , and 3 0 Function), and H : → Fℂ , Hx if x 3 1 0 if 0 ≤ x ≤ 1 . Find m H if 1 1, 2 3 3 , x m H 0 , and m H ℂ 3 . d. Prove that if f : , → is measurable, then f −1 a ∈ ∀a ∈ . You may use any results we proved or stated in class. e. Let E ⊂ 0, 1 be the non-measurable set we constructed in class, and define x if x 0 f: f. g. 6. , → , fx x 1E x 1 x 1 0,1 −E x if x ∈ E 1 if x x if x ∈ 0, 1 − E . Is f measurable? Is 1 f −1 a ∈ ∀a ∈ ? What conclusion can you draw regarding the converse of the statement in problem d? Consider a not-necessarily-measurable function f : , → . Show that the class of −1 A⊂ :f A ∈ is a field. sets ℒ Measurability of functions can be defined on measurable spaces other than , and , M . The more general definition is this: f : X, → Y, Γ is measurable if −1 f A ∈ ∀ A ∈ Γ. The following questions are based on this definition. I. Prove that the definition of Borel measurable function f : , → stated in class is equivalent to the general definition of measurable in the case f: , → , . (Hint: problem f.) II. Provide an example of a function f : a, b, c , ∅, a , b, c , a, b, c → , that is measurable, and a function g : a, b, c , ∅, a , b, c , a, b, c → , that is not measurable. III. Consider the three measurable spaces , , X, , and X, Γ , where ⊂ Γ. Define F f : X, → , : f is measurable , and G g : X, Γ → , : g is measurable . Prove that F ⊂ G. Homework 6, due Monday, February 20th a. Suppose Y : 0, 1 , , m → , Y . What is F Y x ? Find 1 2 −1 −1 P∘Y , ,P ∘ Y 0 , and P ∘ Y −1 ℂ 3 . 3 b. Suppose X : Xb 3 a, b , 2 a,b , P → 1, P b 2, X a 0, and 3 3 1 , P ∘ X −1 , and P ∘ X −1 ℂ 3 ?. 3 , with P a 1. What is F X x ? Find P ∘ X −1 0 0 if x ≤ − 1 2 c. Consider : , ,m → , , 3 if − 1 2 1 0 if 2 x 1 if 1 x≤ 1 2 x≤1 x , and E 1 ,5 . 4 2 0 if 2 ≤ x Compute dm and 0 d. Suppose F E if x if 1 Suppose G Fℂ 1 and Find g. Find E f. 7. 0 and E as in problem c. Compute dm F x dm F . 0 e. dm. x 2 if 0 ≤ x ≤ 1 with 1 and E if x 3 0 if 0 ≤ x ≤ 1 with if 1 and E as in problem c. Compute dm G x dm G . for as in problem c. W for W : ℤ → , , W n sin 4 Homework 7, due Monday, February 27th a. Consider X : a, b, c , 2 a,b,c , P → , X a 1, P c 3 1 . Let 2 n 1, X b 0, X c 3, P a 1, 6 2 1 0,2 4 1 2,5 . Compute dP ∘ X −1 . b. Prove that the condition f ≥ 0 in MCT can be relaxed to f ≥ g, where g . Can it be relaxed even further by simply eliminating the f ≥ 0 condition and not replacing it with anything? 1 c. Consider f n n≥1 : , , m → , , fn x . Compute 2 2 Pb x 1 x2 lim n→ d. f n dm. cos x n2 Consider the Cantor set formed by starting with 0, 1 and removing the middle subinterval of length 4 −k from each remaining interval at step k, k ∈ ℕ. Let the subset of 0, 1 remaining after stage k be A k , and A ∩ k≥1 A k . Demonstrate by direct computation that lim k 1 A dm 1 A dm. Does this result follow from MCT 0,1 0,1 k independent of direct computation? Does it remain true if we phrase it in terms of 1 1 1 A dx? Riemann integrals lim k 1 A dx 0 k 0 No homework due March 6th or 13th. Midterm exam in class on Friday, March 3rd. 8. Homework 8, due Monday, March 20th a. b. 1 0,1/n x for n ≥ 1. For which values of p with 1 ≤ p ≤ is f n n≥1 x a Cauchy sequence in L p , , m ? Suppose 1 ≤ p ≤ q ≤ . Prove that L q 0, 1 , , m ⊂ L p 0, 1 , , m . (Hint: choose Consider f n x c. f ∈ L q and consider x : |f x | ≤ 1 and x : f x 1 separately.) Does your answer change if we replace 0, 1 with ? Consider the measure space x, y , 2 x,y , where x y 1, and the Hilbert space L 2 x, y , 2 x,y , For f, g ∈ L 2 , compute f, g and ||f|| 2 . Compare this to the corresponding undergraduate calculus quantities f, g and ||f|| 2 for two-dimensional vectors in 2 given in terms of their components by f 〈f x , f y and g 〈g x , g y . d. Consider as a one-dimensional L p Banach space. Show that ||f|| p ||f|| 1 for any f ∈ L p and 1 ≤ p ≤ . Prove directly that every absolutely summable series is summable, i.e. take as given that is complete and use it to prove that n≥1 ||f n || p n≥1 f n exists. e. Consider 2 as a two dimensional Hilbert space with the usual L 2 (Euclidean) inner product and norm. Find the projection of v 〈1, 0 on the subspace of 2 spanned by u 〈1, 1 . How much of this carries over to the situation where the vector space is 2 over the field as opposed to 2 over the field ?
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