1. Let (X, M, μ) be a measure space and suppose that {En} is a sequence of measurable set such that X = U1 En and En C En+1, for all ne N. Let f: X → R be a measurable function such that fx|f|du <∞. Show that lim n→+∞ lim n→+∞ √₁₂ |f|dµ = 0. So EC 2. Let g: R → R be a measurable function such that fx|g|dm <∞ (here m is the Lebesgue measure). Show that (*_ \9(x)|dx + [_~_* 19(x)|dx) = 0.

Q: 1.4.10. Prove that the closed interval [0, 1] and the open interval (0,1) have the same cardinality…

A: (Schr¨oder-Bernstein theorem) :Let A and B be sets. Suppose there are injective functions f : A → B…

Q: Let F be the co

A: Here,for G∈F and △∈D={cG1-G2:CER,Gj∈F, j=1,2}. Then, TG+t△-TG=∫G+t△z-ydG+t△ y-∫Gx-ydGy…

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 F: [0, 1] [0, o0] be a measurable function. Assume that lim sup,x " dr 1 then the sequence of…

A:

Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

A:

Q: Let X~ N(2, 9). Compute P(1 ≤ X < 7).

A: From the provided information, Mean (µ) = 2 Variance (σ2) = 9 X~N (2, 9)

Q: # 2. Let S be the set of all even integers, and define a function, f: Z –→ S as follows: f(n) = 2n…

A:

Q: Show that a uniformly continuous function on (a, b) is bounded.

A:

Q: 24. Let f be an arithmetical function (not ne multiplicative arithmetical function g s S fl(k,n)) =…

A:

Q: Let S=X belongs to R: x> =1. a) Show that the set S has lower bound, but no upper bounds. b) Show…

A: The answer is given as follows:

Q: Let f : A → B be a function, {B;}jej a family of subsets of B. Prove that f-'(Njej B;) = Njej…

A: Given that f : A → B is a function. Also given that Bjj∈J is a family of subsets of B . We have to…

Q: Prove that a set E ⊂ ℝ is measurable iff l(I) = m∗(I ∩ E) + m∗(I ∩ E′) for any open interval I = (a,…

A: A set E ⊂ ℝ is measurable if and only if l(I) = m∗(I ∩ E) + m∗(I ∩ E′) for any open interval I =…

Q: Let 'H' and 'Gʻ are two non-empty sets which are defined as; H = (x; -1 sIs1}=G and modulus function…

A:

Q: Let (X, E) be a measurable space and (En) be a sequence of pairwise disjoint sets in E. If, for each…

A: Given X,∑ is a measurable space and En  be a sequence of pairwise disjoint sets in ∑. Given f is…

Q: Px (w) is a continuous function of w in (- ∞, 0) for any real value of ow, and in fact, it is…

A:

Q: 10) Prove that if fεg are continuous functions R on IR then fog is also continuous on i

A: Step 1:To prove that if f and g are continuous functions on R then fog  is also continuous on R, we…

Q: 2.58 Given an example of f E C (a, b) for which the range of f is a) a finite open interval. b) an…

A:

Q: Let f : N → P(N) be the function defined by f(n) = {n, n + 1, . . . , 2n} The following statement…

A: Given a mapping f where f:ℕ→Pℕ defined by  fn=n,n+1,n+2,...2n It can be seen that the set fn…

Q: Provide two disjoint nonempty open sets C, D. Is it possible for a function g : R → R to be…

A: A set is a collection of elements like, numbers, objects. The numbers can be called as set of real…

Q: continous real valued function 58) Let f be a that the set inverse image with respect to f of an on…

A:

Q: f : A → B is a function, X is a finite subset of Z, and k is a positive integer For each of the…

A: Given: f : A → B is a function, X is a finite subset of Z, and k is a positiveinteger. We have to…

Q: Sj. Let f be a measurable function with domain DER and let ECD be a measurable subset of D. Show…

A:

Q: SOLVE STEP BY STEP IN DIGITAL FORMAT ジョッシ ♥♡ A A * C * ! ! ?? !! ? ?! ¿ ¡ !? W X √√√ X X X X DO - B…

A: Given: X is an uncountable set with cofinite topology . To prove : Every continuous function f: X→ℝ…

Q: Let (A,) be a poset. Define a function f: A → P(A) by f(x) = {a € A: a <x}. What can you say about x…

A: Here given that be a poset.Define function is  by Noted that here  is power set of  .Here we have to…

Q: 2. The Mean Value Theorem (M.V.T.). First proved by Joseph-Louis Lagrange. 1.) Let f be a continuous…

A:

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

A:

Q: Prop (1.23 ≤ x ≤ 1.72) = Prop (-2.50 ≤ x ≤ 0.5) = Prop (x ≥ 3.00)

A: here use normal distribution table

Q: Let f be the function given by f(x)=x'. Find all values of c that satisfy the conclusion of the Mean…

A:

Q: 2. Let a € R be given. Define f: R" → R by f(x) = a¹x, x = R¹. Using the € - & definition, show that…

A:

Q: 8. Prove that a function which is uniformly continuous on X is also continuous on X

A:

Q: 4. Let EC Rd be a measurable set such that |E| > 0, and assume that ƒ : E → [-∞, ∞] is a measurable…

A: Since |E| &gt; 0 and E is measurable, we can find a sequence of closed cubes  in Qn∈ℝd such that E…

Q: et F be the collection of

A: To define, T(G)=∫G(z-y)dG(y),G∈F

Q: We say f has an upper bound U on the interval [a, b] if: for all x on [a, b], we have f (x) ≤ U .…

A:

Q: 3. Give an example of a non-Lebesgue measurable function f: IR→ IR such that || is a measurable…

A:

Q: 3. Let f be entire. Suppose that there exist a EC and > 0 such that f(z) & Da, e) for all z E C.…

A: Since f is entire, it is holomorphic on the entire complex plane. Now, we know that f has a root in…

Q: Let f be a continuous, increasing function on [a, b] and let P be a partition -a for all i = 1, 2,…

A:

Q: Let S4be the set of all strings of 0’s and 1’s of length 4. Define F: S4 → Z nonneg such that F(s)…

A: Using the definition of the function given we answer this question. We just have to count the number…

Q: Let f:X→Y be a function where X= { 2, ,3, 4, 5, 6, 7} and Y ={ 0, 1, 2,3, 4} defined as f() =2x(mod…

A:

Q: 12. For each n € N, define An = [0,1 + ½] and B₂ = [0, 1 − 1]. - (a) Prove that Uº₁ An is an…

A:

Q: Let (X, E, µ) be a measure space and let {fn} be a sequence of nonnegative measurable functions on…

A:

Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A→A and f: A → A. (a)…

A: