Let ([0,1], L, m) be a Lebesgue measure space,and let A be a nonempty measurable subset of[0,1]. Let {E}%=1 ≤ [0,1] be a countabledisjoint collection of Lebesgue measurablesets. Let f:[0,1] → (0,1] be a measurablefunction. Show that for every e > 0, there is anatural number N and a set C such that- for all x E CE.11m(C) < € andNE+1NE

Answered: Image /qna-images/answer/bc63928d-c669-408f-9079-1aae68dab1f7.jpg

Answered: Let ([0,1], L, m) be a Lebesgue measure…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 24E: 24. Let, where and are nonempty. Prove that for all subsets and of. Prove that for all subsets...

See similar textbooks

Similar questions

Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
24. Let, where and are nonempty. Prove that for all subsets and of. Prove that for all subsets and of. Give an example where there are subsets and of such that. Prove that for all subsets and of. Give an example where there are subsets and of such that .
Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
Let (X, B) be a measurable space and {μn} a sequence ofmeasures with the property that for every E ∈ B,μn(E) ≤ μn+1(E), n = 1, 2,... .Let μ(E) = limn→∞ μn(E). Show that (X, B, μ) is a measurespace.
Let (X,T) be a topological space, K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø for all n. If K⊇F̅0⊇F̅1⊇....⊇F̅n⊇........ then prove that (by way of contradiction) n=0∩∞ F̅n ≠ø
Let A be a subset of a space X and let x ∈ X. Then x ∈ A iff there exists a net in A (that is a net which takes values in the set A) which converges to x in X
Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.
Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerableset T2.
Prove that if 7 is a constant function on [", $] then C 7DE= F(7, &̇)for any tagged partition &̇ of [", $].
Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the sup-norm. Let T: C[0,1] -> C[0,1] be a bounded linear operator such that T(f) is continuously differentiable for every f in C[0,1]. Prove or disprove the following statement: "If T is injective, then T^{-1} is also bounded."

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS