2. Consider a measure space (X. A. u) and let s be a nonnegative measurable simple function on X. For E E A, define (E) = s du. Prove that o is a measure on A.

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: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

Q: Let X N (u,1) of size n. Obtain UMVUE for M.

A: We will use Lehmann-Scheffe theorem to find the uniformly minimum variance unbiased estimator for μ.

Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

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: In Exercises 1-7, (X,M) is a measurable space. 1. Let f : X R and Y = f-1(R). Then f is measurable…

A: Given: Let's consider X,M is a measurable space.  Here f:X→ℝ¯ and Y=f-1ℝ

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: Consider X = C[0, 1] with the norm = max |x(1)|, IE[0,1] and let T: X - X be the mapping given by…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

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

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: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

A:

Q: If the characteristic function of a RV X is f,(@) and if Y = aX + b, then O,(@) = ibo e

A:

Q: If E and E, are measurable , then (E,UE,) is not measurable. ylgn

A: False.

Q: If the Wronskian W of f and g is 3e^(4t), and f(t) = e^(2t), find g(t)

A: Wronskian W of f and g is given by:

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

A:

Q: is not an endpoint of any of the intervals (e) Prove that eC but in any of the sets A (k e N),

A:

Q: defined BCX,x). a norm Dn

A:

Q: be any nonempty set and let µ* be an outer measure on P(X) and let A, B e P(X). If at ne of A, B is…

A:

Q: Suppose (X, S) is a measurable space, E1, . . . , En are disjoint subsets of X, and c1, . . . , cn…

A: Let's suppose that ϕ:X→R defined by, ϕ=c1XE1+c2XE2+⋯+cnXEn So, this function is well defined as…

Q: If E, and E2 are measurable then (E,UE) is not measurable .

A:

Q: Let c E R and suppose that an → a. Show that can → ca.

A:

Q: (a) Let X = {m, n, p, q}, and = {ø, X, {m}, {n}, {p, q}, {m, n}, {n, p, q}, {m,p, q}}. Is a sigma…

A:

Q: Let x E RP be a cluster point of a set E and that x ¢E. Prove that x is a boundary point of E.

A:

Q: If E, and E2 are measurable , then (E,UE,) is not measurable . صواب İhi

A: Here we have given E1 and E2 are measurable sets

Q: Let s in G, s is not equal to 1.

A:

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

A:

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: Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is…

A:

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

A: Let X,A,μ be a measure space. Let {En}n=1∞⊂A and E1⊆E2⊆....then, To prove: μ∪n=1∞En=limn→∞μ(En).

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

A: Given :-(a, ∞) is not measurable.

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

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: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

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.

Q: Let X be a nonempty set and fix x E X. Define (1, rE A (A) = 10, 4 A

A:

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

A:

Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

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: let X~~ Per(P) Prove ô is unbiare if ô-X let x exp(0) Prave e is unbiar

A: Introduction: One random variable follows Bernoulli with parameter p and other random variable…

Q: (a) Let X = {m,n, p, q}, and = {ø, X, {m},{n}, {p, q}, {m,n},{n,p, q}, {m, p, q} %3D a sigma algebra…

A: Given: Let X=m, n, p, q and Σ=∅, X, m, n, p, q, m, n, n, p, q, m, p, q. (a) Find whether Σ a sigma…

Q: Please solve...

A:

Q: For any x E R, there exists n EN such that x < n; ylgn ihi

A: We have to state whether given statement is true or false.

Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

A:

Q: Let fELoi Show that i for any E>o, the set 3- S has a positive measure -

A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

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: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement of P, then: L(f;P)…

A: Correct option :- B Detailed Explanation :

Q: Let E ⊆ R. Show that E is measurable if and only if χE is measurable

A: