Let A1 {ACR| A is countable} and A2 = {AC R| A° is countable} which is a measurable space? %3D (IR, A1 N A2) (IR, A¡) (IR. 12) (R. 40.42)

Q: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: Let {X;} be i.i.d.~ Exp(0) and a; > 0. Denote by Z = min{a,X1,..., amX,}. Suppose that 0. Prove that…

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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: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

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Q: (a) Let A = { 1.r €R. Show that A is a bounded subset of R. %3D 2x2+1 Hence find inf A and sup A.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 4. Assuming that the c possible subset of R (c) f(x) = V5 –

A: Use the definition of domain and range to solve this problem

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: Given a 2 0 and b > 0, let u = a and v = Vab Use the Cauchy-Schwarz inequality to compare the…

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Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: Let A = {AC R| A is countable} and Az = {A c R| A° is countable} which is a measurable space? (R,…

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Q: prove the ged (atb,a?+b³) =16Y2 assuming that gcd (a,b) =1 •

A: It is given that gcd(a,b)=1

Q: Let A = {AC R| A is countable} and A2 = {ACR| A° is countable} which is a measurable space? (IR, A1)…

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Q: OVESTON Prove that A is inveitine if at0 and at b l Gind The pivos 9 A %3D a a

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Q: - Show that the area of the triangle whose vertices are the [x1 yi det X2 Y2 A - points (x₁, y₁),…

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Q: Is S = {a + b√(2) : a, b ∈ ℤ} dense in ℝ?

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Q: IS T-dist prove o E(T)=0 QvarしT)=て %3D V-2

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Q: Show that X and Y are independent iff E(f o X – go Y)² E(f o X – E(f o X))² for all Borel-measurable…

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Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: Prove that for Vr > 0, s > 0, √√r + √√5 ± √√r + s r, s

A: # As per the guidelines we are entitled to solve one question at a time please resubmit the other…

Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: || I(f) sup {Lp(f)| P is a partition of [a, b]} a

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Q: Exercise 5. Let (E,T) be a measurable space, let (m,)ısisn be n measures on (E,T), and le (a;)ısisn…

A: Let E,τ be a measurable space and mi1≤i≤n be a n measure on E,τ.  Suppose, ai1≤i≤nbe real positive…

Q: 5. Suppose f E L'(R²) is real-valued. Show that there exists a measurable set EC R? such that fdm2.…

A: Given that f∈L1(ℝ2). And E⊆ℝ2. The objective is to show that ∫Efdm2=∫ℝ2/Efdm2.

Q: (a, o) is not measurable. ylgn ihi

A: Given :-(a, ∞) is not measurable.

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: Let an= こn らnf1 a) de ter mine whetber I an] is conver g ent or diver gent b) determine whether

A: Since part a and part b both are same, I am doing part a.

Q: 2. What is the largest possible c in (1) if X=R² and x₁=(1,0), x₂ = (0, 1)? If X = R³ and x₁ = (1,…

A: Given: c To find: a) X=R2b) X=R3

Q: Let A1 {AC R A is countable} and A2 = {Ac R| A° is countable} which is a measurable space? (R, A1)…

A: Union is the correct answer

Q: 8. (a) Suppose f : [a, b] → R is integrable and L(ƒ, P) = U(ƒ, P) for some partition P of [a, b].…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 2. Consider a measure space (X. A. u) and let s be a nonnegative measurable simple function on X.…

A: We have a measure space χ,Α,μ. Where A is a σ-Algebra and μ is a measure. We are given a function…

Q: Let ECR be measurable. Prove that L'(E)n L°(E) C L*(E).

A: Given E⊆ℝ be a measurable set. Let f∈L7E∩L9E. Let λ=716, therefore:…

Q: 23. Let A be a non-empty set of positive numbers, which is ounded above and let B= {xe R: 1/x e A}.…

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Q: 2. Let f and g both be of bounded variation on [a, b]. Prove that this inequality holds. Vrag SV+V

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Q: 9.K Let X = Q y K = {p € Q : 2 < p? < 3} . Show that K is closed in Q, is bounded, and is not…

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Q: (b) Using Venn Diagram to prove that (A U B)e = A° N B° ?

A: To prove the following using Venn diagram A∪Bc=Ac∩Bc First consider the left hand side  Set A

Q: (a) Prove that if f e C (R²) then a (af af %3D

A: This is a problem of the calculus of several variables, higher dimensional calculus.

Q: Suppose the matri x A is 27 X 36 and the rank of A is 14. Find the nullity of A a e dimension of…

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Q: Let F1 and F2 be 4-dimensional flats in R6, and suppose that F, N F, # Ø. What are the possible…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: Find conditions on a, b, c, and d such that

A: Find conditions on a, b, c, and d such that  B=abcd commutes with both 1000 and 0001 For any matrix…

Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

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Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: 1. Let m*(A) = 0. Prove that for every B, m(BU A) = m*(B). %3D 2. Show that m" (A+ y) = m*(A). %3D…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub- parts for…

Q: Nine Starbucks are located in a 3 mile by 3 mile square area in which there is a 1 mile by 1 mile…

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Q: Let A be a non-empty set of positive nunıbers, which is bounded above and let B = {x e R: 1/x e A}.…

A: