ala lo oepse a ad 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m"(B).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![lo ope a d
1. Let m*(A) = 0. Prove that for every B, m(BUA) = m*(B).
2. Show that m*(A+ y) = m*(A).
3. Show that if A is measurable, then for every y E R A+y is
measurable.
4. Show that if A and B are measurable, then A+B is measurable.
5. Suppose that f2(x) is a measurable function. Is it true that f(x)
is measurable? Prove, or give a counterexample. (EXAM)
6. Give an example of a non-measurable function.
7. Let f be a function on R that is continuous a.e. Prove that f is
measurable.
8. Give an example of a bounded not Lebesgue integrable function.
9. Show that for every positive function f on R there is a monotone
increasing sequence of integrable functions {fn} that converges to f
a.e. AM
10. Suppose that f is a measurable function on R that satisfies the
following condition:
Vn eN m( {r E R: |f(x)| > n} ) <
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m {x ER: |f(x)| 2n} ) >
ne N.
Is f integrable?
12. Prove that the space L [0, 1] is complete.
13. Let E be a set of measure 1, and f be a continuous function on
E. Suppose that pi > P2 > 1. What is bigger ||f p, or || fn |lp2?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2Fa32efd32-da55-438c-8483-f00c8929d477%2Fkjnsb5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:lo ope a d
1. Let m*(A) = 0. Prove that for every B, m(BUA) = m*(B).
2. Show that m*(A+ y) = m*(A).
3. Show that if A is measurable, then for every y E R A+y is
measurable.
4. Show that if A and B are measurable, then A+B is measurable.
5. Suppose that f2(x) is a measurable function. Is it true that f(x)
is measurable? Prove, or give a counterexample. (EXAM)
6. Give an example of a non-measurable function.
7. Let f be a function on R that is continuous a.e. Prove that f is
measurable.
8. Give an example of a bounded not Lebesgue integrable function.
9. Show that for every positive function f on R there is a monotone
increasing sequence of integrable functions {fn} that converges to f
a.e. AM
10. Suppose that f is a measurable function on R that satisfies the
following condition:
Vn eN m( {r E R: |f(x)| > n} ) <
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m {x ER: |f(x)| 2n} ) >
ne N.
Is f integrable?
12. Prove that the space L [0, 1] is complete.
13. Let E be a set of measure 1, and f be a continuous function on
E. Suppose that pi > P2 > 1. What is bigger ||f p, or || fn |lp2?
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