ands 1. Let m*(A) = 0. Prove that for every B, m" (BUA) = m" (B). %3D %3D
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
9
![dt ol t no
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 RA+yis
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. AN
10. Suppose that f is a measurable function on R that satisfies the
following condition:
1
Vn eN m( {r E R: |f(x)| > n}) <
n2
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m( {z€R: If(z)| > 따})>
ne N.
Is f integrable? N
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 l, or || fn llp?
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Transcribed Image Text:dt ol t no
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 RA+yis
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. AN
10. Suppose that f is a measurable function on R that satisfies the
following condition:
1
Vn eN m( {r E R: |f(x)| > n}) <
n2
Prove that f is integrable on R.
11. Suppose that f is a measurable function satisfying
m( {z€R: If(z)| > 따})>
ne N.
Is f integrable? N
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 l, or || fn llp?
1
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