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--- math105-s22:hw:hw2 [2022/01/27 14:54]
pzhou
+++ math105-s22:hw:hw2 [2022/02/03 23:00] (current)
pzhou [Lemma 3]
@@ Line 1: / Line 1: @@
 ====== HW2 ======
+Due on gradescope next Friday 6pm. Please also submit on discord sometime around Wednesday.
@@ Line 10: / Line 10: @@
 In the following, the measurability is defined using Def 2 above.
+==== Lemma 0 ====
+(stolen from Tao's grad [[https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf | measure theory book]])
+{{:math105-s22:hw:pasted:20220127-143526.png}}
 ==== Lemma 1 ====
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 Hint: This is a hard one. First prove that  $A$  can be written as countable union of bounded closed subsets, then suffice to prove the claim that any bounded closed (hence compact) subset  $A$  is measurable.
+Hint 2: Given  $A$  a closed bounded subset of  $\R^n$ , for any $\epsilon>0$, we have some open set  $U$  with  $m^*(U) < m^*(A) + \epsilon$ . Then we just need to show that  $m^*(U \RM A) < \epsilon$ . Note  $U\RM A$  is open. Claim: for any closed subset  $K \in U \RM A$ , we have  $m^*(K) + m^*(A) = m^*(A \cup K) \leq m^*(U)$ , hence  $m^*(K) < \epsilon$ . The key point is to show that there exists an increasing sequence of closed subsets  $K_n \In U \RM A$ , such that  $\lim m^*(K_n) = m^*(U \RM A)$ . Try to construct  $K_n$  as finite union of almost disjoint closed boxes, e.g., use the dyadic subdivision trick.
 What does a closed set look like? Say, the cantor set in $[0,1]$?
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-==== Lemma 1 ====
-{{:math105-s22:hw:pasted:20220127-143526.png}}

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