Exercise 5. Let (E,T) be a measurable space, let (m,)ısisn be n measures on (E,T), and le (a;)ısisn be real positive numbers. For A ET, we let m(A) = Ea;m;(A). %3D i=1 Show that m is a measure on (E, T).

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: 2. Let (S, d) be a discrete metric space and p E S. Identify S i(P), S1 (p), S2(p).

A: We shall provide answers for first question only as per the answering guidelines. For other…

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: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

A:

Q: Let G = Z31, S = {1,5, 6, 25, 26, 30}. %3D %3D Prove that Cay(G, S') is normal.

A: It is given that

Q: Let (X, E) be a measurable space. Let py and pz be two measures defined on X. Then 1. p - H2 is a…

A:

Q: Exercise 4.7 Consider n observations for which y = a + bxi, b>0, holds. Show that = 1

A: Given that there are n observations. yi=a+bxi

Q: Let E = [-2, 3) U {−3, π}. Determine the following max D inf D interior of D boundary of D closure…

A: Solution:-

Q: If u (E) =0, then E is measurable.

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: H.W: Let X = {0,1,2, ... ,10} and let A, B be fuzzy sets in X such that %3D A = {x € X is neither…

A:

Q: التاريخ 27 أمر موح let I = (a,b) be an open interval show that - for every positive sM(IDS.

A:

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

A:

Q: Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by…

A:

Q: T C S E

A: Answer is : Yes

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

A: False.

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. ореn b. closed c. clopen d. is…

A: Let (X, T) be a discrete space.

Q: Exercise. The set N has o accumulation points in R with the euclidean metric.

A: Given sets are N and R. N is the set of natural numbers and R is the set of real numbers. So, N is a…

Q: Let (X, E) be a measurable space. Let u1 and u2 be two measures defined on X. Then 1. µi – 42 is a…

A: I have Provided this answer in step-2.

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

A:

Q: Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per…

A: The regression equation is y^=a0+a1x.   Here, y: be the number of medical doctors per 10,000…

Q: Exercise 4.5 Suppose m is Lebesgue measure. Define x + A {x + y : y E A} and cA real number. Show…

A: Given m is a Lebesgue measure.  Define  x+A=x+y:y∈AcA=cy:y∈A, x∈ℝ,c∈ℝ.

Q: Compute F. Oszs V1-x2 –y? Let E be the hemisphere described by and F=Vz2 + x² +y² i + Vz² + x² + y²…

A: By Gauss divergence theorem:∬∂EF·nds=∭Ediv(F)dvhere givenF=x2+y2+z2( i^+j^ +k^…

Q: Yoko does a weekly exercise program consisting of cardiovascular work and weight training. Each…

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: Let (X, E) be a measurable space. Let u, and µz be two measures defined on X. Then 1. H1 - 42 is a…

A:

Q: Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per…

A: (b) Verify the provided data:

Q: If S is a subset of T , then u* (T) su* (S).

A:

Q: Find an example to show that the convexity of S is necessary in Exercise 19. Reference Exercise 19:…

A:

Q: #6

A: We know that a Gamma density function is: So E[X] is:

Q: A shop sells 3 brands of textile: B,, B,, and B, of yarn, each of which is a blend of Orange (0),…

A:

Q: Show that the autocovariance function can be written as γ(s, t) = E[(xs − µs)(xt − µt)] = E(xsxt) −…

A:

Q: Let z = ƒ(x, y), x = g(s, t), and y = h(s, t). Explain how to find ∂z/∂t.

A: Solution:- The given question is  Let z = ƒ(x, y), x = g(s, t), and y = h(s, t). Explain how to find…

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: 8. Let S= I+T*T: H- H, where T is linear and bounded. Show that s1: S(H)– → H exists.

A: Let S=I+T*T : H→H where T is linear and bounded. Since T is a bounded linear operator and…

Q: (X,) is a sequence of independent identically distributed positive RVs, where E(X,) = a and EX,'=b…

A:

Q: Exercise 6. Let E be an infinite non-demumerable set. For every A € P(E), we set m(A) = 0 if A is…

A: Let E be an infinite non-denumerable set. m will be a measure on PE if it is non-negative, which…

Q: If S is a subset of T, then u*(T) su* (S)

A:

Q: Are the following sets linearly independent? Justify your answer. {xª, x*} 1. {In(x°), In r} 2.

A:

Q: 8.Prove the following relationships: a) I(X,Y)=H(X)-HOX/Y)

A: Solution

Q: Consider a unit square with four vertices in a two-dimensional Euclidean space, i.e., {(x1,x2)|x¿ €…

A: As per the question we have a unit square and we have to choose two points P,Q inside it uniformly…

Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

Q: 7. Which of the following sets are open, and which are closed? Explain your answers by sketching the…

A: A set is open if all points of this set are interior points. A point x belongs to a set A  is said…

Q: Identify the disčontinuities D in the (x, t)-diagrams below. Indicate if they are physically…

A: In the given figure there are three (x,t)-diagram .we have to identify the discontinuity D in…

Q: Exercise 6. Prove that if f [a, b] : interval. → R is continuous, then f([a, b]) is a closed and…

A: 6. Given that f : a,b→ℝ is a continuous function. To find: f a,b is closed and bounded interval.…

Q: Iw (X, D) discrete space A = {2,3,4s A A Bd(A), Euct CA)? EXercise %3D Fined

A: Given that A=2,3,4 The aim is to find A,A°, Bd(A), Eucd(A) A=2,3,4 And, A°=∅ Here A is closure of 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>0:μx:fx>M=0For M>0, defining the set AfM=X:1 fx>M⇒For f∈L∞,…

Q: Exercise 3.6 Prove that if (X, A, µ) is a measure space, B e A, and we define v(A) = µ(AN B) for A E…

A: Given X,A,μ is a measure space and B∈A.  Let X  be a set and A be a σ-algebra on it. A measure ν is…

Q: (b) Let (X, >, ) be a measure space. For A, Be ). Show that μ(AU B ) - μ(Α) + μ(Β) -μ(An ). %3D

A: Since you have asked multiple question, we will solve any one question for you. If you want any…

Q: Exercise 4.11 Suppose m is Lebesgue measure and A is a Borel measurable subset of R with m(A) > 0.…

A: Given, m is Lebesgue measure and A is a Boral measurable subset ofℝ with m(A)>0.  To prove that…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.