(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample. (a) Let Ω = [4, 101 and let A = 16,...
10. Let 0 1 1 A=101 100 . Prove that 10 0 11 Hint: Write A in the form A=9I+ 0 0 0 1 0 0 0 1 0 and observe that To 101 Al=eexploit 1000 /
Problem 3 (7 points) Define three functions A,A,As as follows: βί(x) = 0 whenever x < 0 and A(x)-1 whenever x > 0, Moreover we let A (0)-0, β2(0)-1 and A3(0) . Let f be a bounded function on [-1, 1] (a) Prove that f ER(B1) if and only if lim0+ f(x)-f(0). In this case prove that 1 『(0) elf, -1 (b) State and prove a similar result for A. (c) Prove that f ER(B3) if and only if f...
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by epsilon-delta language, that f is continuous at c in an element of (0, infinity). 2. Let f(0) = 25 with domain (0,00). Prove, by the e-8 language, that f is continuous at CE (0,0)
Question 5 (1 point) S2x4, Let f(2) - <x< 0 5 sin(x), 0 < x < Evaluate the definite integral [ f(x) f(x)dx. 5 O + 10 873 - 10 O 1/25 - 10
QUESTION 16 Let X={1,2,3,4} and T={0,X,{1,2}, {3,4}}. Let f: (X,T) → (X,T) defined by f(1) = 3 , f(2)= 1, f(3) = 4 ,f(4) = 2. Then f is continuous at 2. True False QUESTION 12 The function f(x) = xis open. True False QUESTION 6 Let f: X→ Y be a continuous function and A be a path connected in X, then f(A) is connected in Y. True False
3. Let f, g : a, bl → R be functions such that f is integrable, g is continuous. and g(x) >0 for al x E [a, b]. Since both f,g are bounded, let K> 0 be such that f(x)| 〈 K and g(x)-K for all x E la,b] (a) Let η 〉 0 be given. Prove that there is a partition P of a,b] such that for all i (b) Let P be a partition as in (a). Prove...
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
(3). Let F be a field and let f(x) = ao-chx +-.. + an-,Kn-1 + an&n E F[x]. Prove that x - 1 is a factor of f(x) if and only if ao+ aan 0
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...