Define u* on the power set of R by Jo if A is bounded, 4*(A) = 1 if A is unbounded. Is u* an outer measure or not? (Prove your claim.)

Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

Q: Prove: If S is compact, and ƒ is continuous on S, then ƒ takes a minimum value some- where in S.

A:

Q: Exercise 3.3.7: Prove Corollary 3.3.12: Suppose f: [a,b] → R is a continuous function. Prove that…

A: We are to prove that the direct image f([a, b]) is a closed and bounded interval or a single number.

Q: Let f: [a, b] → R be a bounded function, and let P be a partition of [a, b]. Let P be a refinement…

A:

Q: Let SCR be NON empty. Show that ut R is upper bound of 5 if and only if the conditions tER and tu…

A:

Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

A:

Q: Let f: [a, b] -> R be continuous except on a finite number of points in (a, b]. Is f Lebesgue…

A:

Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

A:

Q: (N, F, p) (S,F) is a measure space and is a measurable space. y : N → Ñ ₂ F – Ē - measurable…

A:

Q: S (-1,3] U [5, 6) Ø {x ln(x) x>0} inf (S) sup(S)

A:

Q: Let f : E → R be continuous such that |f(x)|≤|x|, for all x ∈ E. Is f uniformly continuous, justify

A: According to the given information,

Q: Letf:E→R be uniformly continuous. (a) Prove that E is bounded⇒f(E) is bounded.

A: Given, f:E→ℝ is uniformly continuous. Since, E is a bounded subset of ℝ and f is uniformly…

Q: Let f:R→R. If f is defined b then it is Cantindous. foo=2x+1 if X>o if X5o

A:

Q: 1. Let (X, M, µ) be a measure space, fn : X → [0, 1] a sequence of measurable func- tions. Show that…

A:

Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

A: According to the given information, it is required to explain that:

Q: Exercise 1. Let f : A → B be a function with X a subset of A. Prove that if f(X) = 0, then X = 0.

A:

Q: For any k > 0 E Z, show that there is a function f : Z* → Z which is both one-to-one and onto. (Here…

A:

Q: Evaluate both integrals of the Divergence Theorem for the following vector field and region. Check…

A: Given: There is a vector field F= (z-y, x, -x) and a region D. To Evaluate: One has to find the…

Q: Exercise 2.23: Let X be a nonempty set. Define u*: P(X):→→ Ras: [0 If A is finite μ* (A) =- 1 If A…

A:

Q: Exercise 3.2.7: Prove the following statement. Let SCR and ACS. Let f: S→R be a continuous function.…

A:

Q: Prove that the Frobenius norm is consistent. That is, for A, B € Rxn, show that ||AB|| ≤ ||A||||B||F

A:

Q: Let A(0, π) υ (π, e 1| Find the limit point(s) of A. а. b. Is A an open set?

A: Given set is A = 0,π ∪ (π,e2]

Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x&gt;0 is bounded. To prove that the set x∈R:x2−25x&gt;0 is unbounded.

Q: 2. Let (X, A, μ) be a measure space. If {E} CA and E₁ E₂ E3 ..., then prove **** that (UE)= lim…

A: A limit is a value that a function approaches as the input approaches some value. Limits are…

Q: Exercise 2.35. Suppose X is a topological space, and for every pe X there exists a continuous…

A: It is given that X is a topological space. For every element p∈X there exists a continuous function…

Q: Ja (16) Let f e Lla, b]. Show that if g is a bounded measurable function then fa E Cla bl

A:

Q: Q. Consider the measure space (R, M(R), µ), if x € Q 1 -2 let k(x) = { , h(x) = -2,then %3D ¢ Q if x…

A:

Q: 7. Prove that If and g are bounded Lipschitz continuous on A, then the product h = fg is Lipshitz…

A:

Q: Let N be a nonempty set and B a o-ring on N. Let µ, v be two measures on B and μ(A) < ν(Α), A € B.…

A: Let Ω be a non- empty set. And B a sigma ring on Ω. Let μ,ν be the two measures on B And ΩA≤νA ,…

Q: 1 Suppose f: R→ R has the rule f(x) = 21 +1 Select one: O A. fis 1 - 1 but not onto R O B. f is…

A:

Q: Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

A: Suppose   is convex. We have to show that:

Q: Suppose S, T ∈ ?(V) are such that ST = TS. Prove that range S is invariant under T.

A:

Q: Exercise 10. Show that if ACB, then ACB.

A:

Q: (a) Assume that f: A onto B and g: B → A. Prove that g = f-¹ iff g of = IA or fog = IB (Theorem…

A: As per the policy we are solving question (a) for you .For the solution of (b) please repost the…

Q: Show using Definition 4.3.1(Continuity) that if c is an isolated point ofA ⊆ R, then f : A → R is…

A: If c∈A⊆ R is an isolated point of A ⊆ R. To prove  f : A → R is continuous at c. Let ∈&gt;0 , Since…

Q: Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is…

A:

Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

Q: Let f : Rn → R be convex with dom(f) = Ω ⊂ Rn an open set. Show that f is a continuous function on Ω

A: Concavity and convexity are two important analytical natures of a real-valued function. A function…

Q: 3.17 Let A be a subset of R, and let f be a bounded function with domain A. Show that, if BC A, then…

A: Given that A,B are subsets of ℝ andB⊂A. As f is a bounded function in A and B⊂A then fB⊂fA

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…

A: