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ands 1. Let m*(A) = 0. Prove that for every B, m" (BUA) = m" (B). %3D %3D

ands 1. Let m*(A) = 0. Prove that for every B, m" (BUA) = m" (B). %3D %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.2: Norms And Distance Functions
Problem 43EQ
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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
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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