4 (n)4) Let E be measurable and f: E→ R be measurable. Let y=k/n for ke Z andne N. Let Ik =)) so that clearly R = (). Let E() = f((n)) and define(n)00(n)On(x) = x()(x).k=-∞Prove that each on is measurable and that on →f uniformly on R.Hint:=For the former, if on,N(x) Σk-N-1X(n) (x) then prove that we havelim No On,Non pointwise on E. Why is each on,N measurable?4

xStep 1: To prove that each (ϕn) is measurable and that (ϕn) converges uniformly to (f) on (E), we…

Answered: (n) 4) Let E be measurable and f: E→ R…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 25E: 25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove...

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Prove the conjecture made in the previous exercise.
25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove that. Prove that if.
For each n E N, let fn → R be a (F,B(R))- measurable function. (i) Then, each of the functions supnen fn, infnen fn, lim supn→∞o fn, and lim infn oo fn is (F, B(R))-measurable. (ii) The set A = {w : limn→∞o fn(w) exists and is finite} lies in F and the function h = (limn→∞ fn) · IA is (F, B(R))-measurable.
For a derived varaible f : states → R is strictly decreasing if and only if for states q, q' : q → ď = f(d) > f(q) O q + d = f(d) > f(q) O q → ɖ = f(d) f(q)
If fn → f uniformly and the functions fn have only jump discon- tinuities, prove that f has only jump discontinuities.
4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C T. (b) Under what condition does equality hold?
to it 27. Let f(x) = [x2/3]. Find /(S) if a) S-2, 1, 0, 1, 2, 3). b) S (0, 1, 2, 3, 4, 5). c) S = (1, 5, 7, 11). d) S (2, 6, 10, 14).
Unif (-1,1). Compute (a) E(X), (b) var(X), (c) P(-2 0.5). Exercise 1. Suppose that X N
14. Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing. Assume next that f"(x) exists for every x € (a, b), and prove that f is convex if and only if f"(x) 20 for all x e (a, b).
2. Let (r) be the function defined by V(x) = 1. Let 1+x In(x) Vovo...(x) n-copies Note that alternate definitions proceed by induction: Let V₁(x) In (V(x)) or, equivalently, define ¥n+1 = (In(x)). = (x) and define Vn+1 a) What is the domain of Vn(x) for n = {1,2,3,4}? b) What is your guess for the domain of Vn in general? c) Calculate (x) for n € {1,2,3,4}. Arrange your calculation in a way to make it easily under- standable. There are various approaches here and some are easier to understand than others. Do not be afraid to start on one approach and give it up in favour of another. d) What is your guess for (r) in general. You are not asked to prove this by mathematical induction, but it is a good exercise to give it a try.
31. Let f(x) = [x²/3]. Find f(S) if a) S = {-2,-1, 0, 1, 2, 3). b) S = c) S = d) S = {0, 1, 2, 3, 4, 5). {1, 5, 7, 11). {2, 6, 10, 14).
2. Let A = {1, 2, 3, ..., 10}. Consider f:P(A) → N given by f(B) = |B|. That is, f takes B subset of A to |B|- b. Is f surjective? b. Find f-'(3).

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