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Ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B).

Ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 9BEXP
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oe lion.0 moonl e of a ovco ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B). NT po w eudoeco o in T 1 2bo 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. 10. Suppose that f is a measurable function on R that satisfies the following condition: 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({r €R: |f(x)| 2 n}) 2 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 ƒ be a continuous function on E. Suppose that pi > P2 > 1. What is bigger || f l, or || fn |lp? 1
Transcribed Image Text:oe lion.0 moonl e of a ovco ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B). NT po w eudoeco o in T 1 2bo 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. 10. Suppose that f is a measurable function on R that satisfies the following condition: 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({r €R: |f(x)| 2 n}) 2 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 ƒ be a continuous function on E. Suppose that pi > P2 > 1. What is bigger || f l, or || fn |lp? 1
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