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: V [2] Let u* be an outer measure on a set X such that B is a subset of X, then u*(AU B) = u*(B).…

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

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Q: If A is a bounded normal operator

A: Given, if A is a bounded normal operator on a Hilbert space H then prove that A*x=Ax for all x∈H

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

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Q: Let M be the set of all measurable sets and m is the Lebesgue measure on R. Show that if A, B E M…

A: First we prove a and b by using simple rules of sets and measure. Then we generalise the results for…

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

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

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

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

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

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Q: Let 1 € R be an Euclidean domain with the strictly positive Euclidean norm p : R \ {0} → N. Assume…

A: Given that 1∈R, where R is an Euclidean domain. By given we have φab=φaφb. We prove that if a is an…

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

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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.

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Q: c. Determine the truth value of each of these statements if the universe of discourse of each…

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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…

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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…

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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: Let X be a non-empty set and F be an arbitrary σ-algebra on X. Define a function μ : F → [0, ∞] by…

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Q: Let a > 0. Show that if A C R is bounded and B = {ax : x € A}, then inf(B) = a(inf(A)).

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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: Let (X, E, μ) be a finite measure space. We write An →A as n → if and only if 1 An (x) → 1A (x) as n…

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Q: Suppose S is a o-algebra on a set X and AC X. Let SA = {E ES: ACE or AnE=Ø}. (a) Prove that SA is a…

A: Ans-We are given two statements below which we have to prove. Suppose S is a σ-algebra on a set X…

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

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Q: Let N be a measurable set of finite measure and let 1 <p< q < ∞∞. Show that if fn f in Lª(N), then…

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

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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…

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Q: 6. Use the definition to show: if fn →f uniformly on S and 9n → g uniformly on S, then fn +9n → f +…

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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: • Graph f (x) = x° + 3x² + 3x + 3 using -2, and the interval -2, –1.7], find each of the following…

A: There is a change in every function. When the variable moves, it is obvious that the function also…

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

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Q: 4. Define f:R → R, define f (x) = 3x + 5 for all x e R. Show that a. fis one - to - one; b. f maps R…

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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…

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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: On the measurable space (R, B (R)), let u: B (R)→ R be defined as: μ(A) = The number of all rational…

A: Given: We are given a measurable space , where represents the collection of Borel sets on the real…

Q: Exercise 1.2A Identify the next shape in the series. 1. 2. 米 | ? | |3. -

A: We want to look for patterns.  Part(1) Let us first see the shape outside.  The first figure is…

Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.