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