Question 1. Give an example of a complete metric space (X, d) and a function f...
8. A subsetD of a metric space X is dense if for all E X and all e E R+ there is an element yE D such that d(x, y) <. Show that if all Cauchy sequences (yn) from a dense set D converge in X, then X is complete.
In the following, (X,d) is an arbitrary metric space and (X,d,μ) is an arbitrary metric measure space. (6) Recall the definition of bounded set: The set A C (X, d) is bounded if δ(A) < 00 where 6(A)p d(a,a). (X,d) with ACBand B is bounded then A is bounded (a) Show that if A, B (b) Fix a set A. I B - (r), a single point, show that D(A, B)-0 if and only f (c) Prove that the function...
(8) Give an example of a metric space (X, d), a o-algebra A in X, and a continuous function f X ->R which is not measurable with respect to A. (Does this question make sense if (X, A) is a measurable space without a metric?) (8) Give an example of a metric space (X, d), a o-algebra A in X, and a continuous function f X ->R which is not measurable with respect to A. (Does this question make sense...
A subset D of a metric space (X, d) is dense if every member of X is a limit of a sequence of elements from D. Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense subset of X. 1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
(b) Find an example of an open set G in a metric space X and a closed subset F of G such that there is no δ > 0 with {x : dist(x, F) < δ} C G
(1 point) The joint probability density function of X and Y is given by f(x, y) = cx – 16 c”, - <x< 0 < b < co alt 0 < y < 0 Find c and the expected value of X: c = E(X) =
0, P - 0 2 ) - 1. 3. Let (X, A,) be a complete measure space. Assume that A, B E A with (A) = (B) < . Show that if A CCCB, then CE A. 4 Let A and Rhe two collections of euheete of Y Aceume that any cot in 4 halanes
3. Let f : (a,b) +R be a function such that for all x, y € (a, b) and all t € (0.1) we have (tx + (1 - t)y)<tf(x) + (1 - t)f(y). Prove that f is continuous on (a,b).
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
Q3 * You are given two subsets A, B C M of a metric space M. Define p by p=inf{d(x, y)| 2 € A, Y E B}. Prove that, if p > 0 then A and B are separated. Give an example where p=0, but A and B are not separated. Q4 Show that Q as a subset of R is disconnected. Likewise show R Q is disconnected.