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g I OPn oi lo ponnipes in b 0Nat hol tol novun and 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m (B). %3D

g I OPn oi lo ponnipes in b 0Nat hol tol novun and 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m (B). %3D

Advanced Engineering Mathematics
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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g I 0 m aot lo ponaipe a od A a ole bs 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B). vo A 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(r) 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 ER: |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')| > n} ) > ÷, 1 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 lp, or || fn ||lp?
Transcribed Image Text:g I 0 m aot lo ponaipe a od A a ole bs 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B). vo A 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(r) 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 ER: |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')| > n} ) > ÷, 1 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 lp, or || fn ||lp?
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