12, 2024 NOTE: Throughout m* denotes Lebesgue outer measure and m denotes Lebesgue measure. Also, I will grade all five problems but only count the best four! 1) Let A CR satisfy m*(A) < ∞o. Prove that there exists a Gs set G where m*(A) = m*(G). Hint: Look at m*(A)+1/n for every n E N and use the definition of inf and m* to obtain an open set Gn where AC Gn and m* (Gn) < m*(A) + 1/n. The way to define the G should now be pretty obvious (and the fact G is Gs should be immediate). 1
12, 2024 NOTE: Throughout m* denotes Lebesgue outer measure and m denotes Lebesgue measure. Also, I will grade all five problems but only count the best four! 1) Let A CR satisfy m*(A) < ∞o. Prove that there exists a Gs set G where m*(A) = m*(G). Hint: Look at m*(A)+1/n for every n E N and use the definition of inf and m* to obtain an open set Gn where AC Gn and m* (Gn) < m*(A) + 1/n. The way to define the G should now be pretty obvious (and the fact G is Gs should be immediate). 1
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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Transcribed Image Text:12, 2024
NOTE: Throughout m* denotes Lebesgue outer measure and m denotes Lebesgue measure.
Also, I will grade all five problems but only count the best four!
1) Let A CR satisfy m*(A) < ∞o. Prove that there exists a Gs set G where m*(A) =
m*(G).
Hint: Look at m*(A)+1/n for every n E N and use the definition of inf and m* to obtain
an open set Gn where AC Gn and m* (Gn) < m*(A) + 1/n. The way to define the G should
now be pretty obvious (and the fact G is Gs should be immediate).
1
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