Let (X, B, μ) be a measure space. Suppose Y ∈ B. LetBY consist of those sets in B that are contained in Y andμY (E) = μ(E) if E ∈ BY . Show that (Y, BY , μY ) is ameasure space. 7) Let (X, B, H) be a measure space. Suppose Y E B. LetBy consist of those sets in B that are contained in Y andHY (E) = (E) if E E By. Show that (Y, By HY) is a%3Dmeasure space.

Answered: Image /qna-images/answer/e2247796-cd14-4207-8206-e8461c1b8175.jpg

Answered: )Let (X, B, p) be a measure space.…

)Let (X, B, p) be a measure space. Suppose YE B. Let By consist of those sets in B that are contained in Y and HY (E) = p(E) if E E By. Show that (Y, By py) is a measure space.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that, X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: Tx) and (X, Ty) be two topological spaces. Let X-{a, b, c, d}. Tx-{4, X, {a}, {a, b}, {a, b, c}} Y…

Q: Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

Q: LxercisE Let (X,2,4) be a mensure spsce. Let 1 2- (0, 4oo and pn : 2(0,+o0] be two menaures. Prove…

Q: Let (X, dx) and (Y, dy) be metric spaces, and let A and B be disjoint closed subsets of X X Y. Show…

A: Given:Metric spaces and Disjoint closed subsets and Objective: To show that there exists a…

Q: (f) The set B = {(z, y) € R? : ² + y? < 1} as a subset of R? with indiscrete metric is disconnected.…

A: As per guidelines we have to attempt 1st three parts

Q: Let E be a subset of X, where X is a metric space. Show the limit points of E are the same as the…

A: In mathematics, the closure of a subset S of points in a topological space consists of all points in…

Q: Determine, with justification, whether the series converges absolutely, converges conditionally, or…

Q: Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is…

A: Let X be a measure space with σ-algebra M and the outer measure μ. To prove that X is complete. That…

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

Q: ► Exercise 3.7. Suppose X is a topological space and UC SCX. (a) Show that the closure of U in S is…

Q: d T = {0, {a}, {a, b}, {c,d}, {a, c, d}, {a,b, c, d}, X}. (1) Is this space To? Why or why not? (2)…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology. We claim that Z is NOT…

Q: ► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a…

Q: Let X and Y be normed spaces. Then show that the following hold : (i) If Y is a Banach space, then…

A: We consider an arbitrary Cauchy sequence (Tn)n≥1 in BL(X, Y) and show that (Tn)n≥1 converges to an…

Q: Prdlem 6.36 Pert B)

A: It is double Integral

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

Q: let (x,d) be a metric space. define a flow on (x,d)

A: Step 1:The concept of flow is often used in the study of dynamical systems, where X represents the…

Q: The set [0, 1] with the discrete metric is compact. O True O False

Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

Q: If E is a subset of a metric space (X, d), show that E is nowhere-dense in X if and only if E c is…

A: As per the question we are given a metric space (X, d) and E is a subset f X within that metric…

Q: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

Q: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

A: Given topological space X such that X contains an infinite closed discrete subspace. We now show…

Q: Question 4. Let f:X→Y be a mapping from a topological space (X,7x) a topological space (Y,ry). Prove…

Q: Let (x, e) and (y, d) be metric space with e(x, y) = { 1 and let fr→y be a function. Is f…

Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

Q: Let (S,F,u) be a mee e B₁, B2 E F satisfying (E 1) = μ(B₂) for any pair of B₁) = 0. Then ao-algebra…

Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…

Q: Let (Ω,A) be a measurable space and ν1 and ν2 are finite measures on (Ω,A). Prove or disprove that…

A: Given::- Let (Ω,A) be a measurable space and ν1 and ν2 are finite measures on (Ω,A).

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function f(x) =…

Q: 10. Let X be a compact space and let Y be a Hausdorff space. Prove that X the projection Ty : X × Y…

A: We are given that X is a compact space and Y is a Hausdorff space. We need to prove that the…

Q: Let (X, B(R), µ) be a Borel measure space. Suppose that {An = [In-1 12n+1]}, be a family of elements…

A: Given, An=7n-n,12n+1nn=1∞ We need to find the value of μ=Un=1∞n+1n,4n-1nn=1∞

Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology. Here B=A∪frA To prove for…

Q: Theorem (Axioms of a Closure): Let (M,d) be a metric space and let S,T C M. Then (1) 0 = 0 and M =…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: Theorem (Axioms of a Closure): Let (M,d) be a metric space and let S,T C M. Then (1) = 0 and M = M.…

A: Let M , d be a metric space and let S , T ⊆ M. We know the definition of closure of set, If A is…

Question

Let (X, B, μ) be a measure space. Suppose Y ∈ B. Let
BY consist of those sets in B that are contained in Y and
μY (E) = μ(E) if E ∈ BY . Show that (Y, BY , μY ) is a
measure space.

7) Let (X, B, H) be a measure space. Suppose Y E B. Let
By consist of those sets in B that are contained in Y and
HY (E) = (E) if E E By. Show that (Y, By HY) is a
%3D
measure space.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Exercise 3.45. Let X be any space and Y be a discrete space. Show that the Cartesian product X X Y is equal to the disjoint union Lyey X, and the product topology is the same as the disjoint union topology.
2. Let X be a topological space and let CX be the cone on X. (a) Show that CX is contractible. (b) Show that CX is locally connected if and only if X is locally connected.
Let (X, T) and (Y, T₁) be two topological spaces and let f be a continuous mapping of X into Y. None of the choices. If f is onto and (Y,T_1) is a Hausdorff space then (X,T) is Hausdorff. If f is one-to-one and (Y,T_1 ) is a Hausdorff space then (X,T) is Hausdorff. If (Y,T_1) is a T_1 space then (X,T) is a T_1 space. O
Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where d(x, y) = |x – y|, is separable. -
(a) Let (X, d) be a metric space and let Y be a compact subset of X. Show that Y is closed and bounded. (b) Let X be an infinite set with the discrete metric d(x, y) = = 1, if x #y, 0. if x = y. 12 Prove that any infinite subset Y CX is closed and bounded, but not compact.
fcilien be a topological space X then Let collection of path components of Uc; =X. i=1
Specify the boundary and interior of the set S in 3-Space whose points (x,y,z) satisfy the given conditions. Is S open, closed or neither? x>=0, y>0, z<2
Let (X,d) be a metric space and d*(x,y) = d(x,y)/1+d(x,y) Prove that the family of open sets with respect to the metric d is same as the family of open sets with respect to the metric d*.