Exercise 4.6 Let m be Lebesgue measure. Suppose for each n, An is a Lebesgue measurable subset of [0, 1]. Let B consist of those points x that are in infinitely many of the An. (1) Show B is Lebesgue measurable. (2) If m(An) > 8 > 0 for each n, show m(B) > 8. (3) If E1 m(An) < o, prove that m(B) = 0. (4) Give an example where , m(An) = ∞, but m(B) = 0. n=1
Exercise 4.6 Let m be Lebesgue measure. Suppose for each n, An is a Lebesgue measurable subset of [0, 1]. Let B consist of those points x that are in infinitely many of the An. (1) Show B is Lebesgue measurable. (2) If m(An) > 8 > 0 for each n, show m(B) > 8. (3) If E1 m(An) < o, prove that m(B) = 0. (4) Give an example where , m(An) = ∞, but m(B) = 0. n=1
Advanced Engineering Mathematics
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![**Exercise 4.6**
Let \( m \) be Lebesgue measure. Suppose for each \( n \), \( A_n \) is a Lebesgue measurable subset of \([0, 1]\). Let \( B \) consist of those points \( x \) that are in infinitely many of the \( A_n \).
1. Show \( B \) is Lebesgue measurable.
2. If \( m(A_n) > \delta > 0 \) for each \( n \), show \( m(B) \geq \delta \).
3. If \( \sum_{n=1}^{\infty} m(A_n) < \infty \), prove that \( m(B) = 0 \).
4. Give an example where \( \sum_{n=1}^{\infty} m(A_n) = \infty \), but \( m(B) = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2Fa91b5cc4-98ee-41b1-9e55-0a4791ed66f8%2F5tid66_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 4.6**
Let \( m \) be Lebesgue measure. Suppose for each \( n \), \( A_n \) is a Lebesgue measurable subset of \([0, 1]\). Let \( B \) consist of those points \( x \) that are in infinitely many of the \( A_n \).
1. Show \( B \) is Lebesgue measurable.
2. If \( m(A_n) > \delta > 0 \) for each \( n \), show \( m(B) \geq \delta \).
3. If \( \sum_{n=1}^{\infty} m(A_n) < \infty \), prove that \( m(B) = 0 \).
4. Give an example where \( \sum_{n=1}^{\infty} m(A_n) = \infty \), but \( m(B) = 0 \).
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