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 3. Show that if A is measurable, then for every y ER A+y is measurable.
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 3. Show that if A is measurable, then for every y ER A+y is measurable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
11
![of 9.a o o h
1. Let m*(A) = 0. Prove that for
every B.
m* (BU A) = m*(B).
ovo
2. Show that m* (A+y) = m*(A).
3. Show that if A is measurable, then for every y ER 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 ƒ 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:
1
Vn eN m {r E R: |f(x)| 2 n} ) <
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m {x €R: |f(x)| > n} ) >
ne N.
Is f integrable?
12. Prove that the space L(0, 1] is complete.
13. Let E be a set of measure 1, andf be a continuous function on
E. Suppose that pi > P2 > 1. What is bigger || f llp, or || fn lp?
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Transcribed Image Text:of 9.a o o h
1. Let m*(A) = 0. Prove that for
every B.
m* (BU A) = m*(B).
ovo
2. Show that m* (A+y) = m*(A).
3. Show that if A is measurable, then for every y ER 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 ƒ 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:
1
Vn eN m {r E R: |f(x)| 2 n} ) <
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m {x €R: |f(x)| > n} ) >
ne N.
Is f integrable?
12. Prove that the space L(0, 1] is complete.
13. Let E be a set of measure 1, andf be a continuous function on
E. Suppose that pi > P2 > 1. What is bigger || f llp, or || fn lp?
1
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