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Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete. Let M be a ơ-algebra of sub...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Let (X,A,μ) be a metric space. 4. Let A and B be two collections of subsets...
Let (X,A,μ) be a metric space.
4. Let A and B be two collections of subsets of X. Assume that any set in A belongs to o(B) and that any set in B belongs to O(A). Show that o(A) = 0(B).
2. Let S-{a,b,c,d) and let F1, F2 be ơ-algebras of subsets of S2 given by a....
2. Let S-{a,b,c,d) and let F1, F2 be ơ-algebras of subsets of S2 given by a. Is FînF, a ơ-algebras of subsets of S2? Why (or why not)? b. Is F1 UF, a ơ-algebra of subsets of O? Why(or why not)? c. What is cardinality of 2 ( denoted by #(29) or 12 l). d. Find the Power set of (denoted by 2 ).
1. Let (X, d) be a metric space, and U, V, W CX subsets of X....
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is...
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is a compact subset of X, and Z C Y, then Z is compact if and only if Z is closed (b) IfY. Y are a finite collection of compact subsets of X, then their union Y1 U...UYn is also compact. (c) Every finite subset of X (including the empty set) is compact.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular. (3) Let X be a loc...
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
Problem 1.11. Let P be a probability measure on R, equipped with the Borel ơ-algebra. Let...
Problem 1.11. Let P be a probability measure on R, equipped with the Borel ơ-algebra. Let F(x)-P((-00,2]). P rove that f is non-decreasing right-continuous, F(x) → 0 as x →-00, and F(x) → 1 as x → oo. Prove that if P and Q are two probability measures such that P((-oo, x Q((-00,x]) for all x rational, then P , ie. P(A) = Q(A) for any Borel- measurable set A.
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Could anyone help me with the question (d) and (e)? I've finished the question (a), (b) and (c). You don't need to solve the question (a), (b), and (c), and you could use them directly. And...
Could anyone help me with the question (d) and (e)? I've
finished the question (a), (b) and (c).
You don't need to solve the question (a), (b), and (c),
and you could use them directly.
And the following 2 are (b) and (c).
(a) Let (X,M, μ) be a measure space and T : X → y a mapping from X oni at Y Prove thai (i) N (B C y : T-1 (B) EM} is a σ-algebra of subsets...
4) Let D be the set of all finite subsets of positive integers. Define a function...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Let (X,A,μ) be a metric space.
4. Let A and B be two collections of subsets of X. Assume that any set in A belongs to o(B) and that any set in B belongs to O(A). Show that o(A) = 0(B).
2. Let S-{a,b,c,d) and let F1, F2 be ơ-algebras of subsets of S2 given by a. Is FînF, a ơ-algebras of subsets of S2? Why (or why not)? b. Is F1 UF, a ơ-algebra of subsets of O? Why(or why not)? c. What is cardinality of 2 ( denoted by #(29) or 12 l). d. Find the Power set of (denoted by 2 ).
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is a compact subset of X, and Z C Y, then Z is compact if and only if Z is closed (b) IfY. Y are a finite collection of compact subsets of X, then their union Y1 U...UYn is also compact. (c) Every finite subset of X (including the empty set) is compact.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
Problem 1.11. Let P be a probability measure on R, equipped with the Borel ơ-algebra. Let F(x)-P((-00,2]). P rove that f is non-decreasing right-continuous, F(x) → 0 as x →-00, and F(x) → 1 as x → oo. Prove that if P and Q are two probability measures such that P((-oo, x Q((-00,x]) for all x rational, then P , ie. P(A) = Q(A) for any Borel- measurable set A.
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Could anyone help me with the question (d) and (e)? I've
finished the question (a), (b) and (c).
You don't need to solve the question (a), (b), and (c),
and you could use them directly.
And the following 2 are (b) and (c).
(a) Let (X,M, μ) be a measure space and T : X → y a mapping from X oni at Y Prove thai (i) N (B C y : T-1 (B) EM} is a σ-algebra of subsets...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
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