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math105-s22:hw:hw2

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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]
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 ====== HW2 ====== ====== HW2 ======
 +Due on gradescope next Friday 6pm. Please also submit on discord sometime around Wednesday. 
  
  
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 In the following, the measurability is defined using Def 2 above.  In the following, the measurability is defined using Def 2 above. 
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 +==== Lemma 0 ====
 +(stolen from Tao's grad [[https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf | measure theory book]])
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 +{{:math105-s22:hw:pasted:20220127-143526.png}}
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 ==== Lemma 1 ==== ==== 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: 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. 
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 +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 $[0,1]$?  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}} 
  
math105-s22/hw/hw2.1643324073.txt.gz · Last modified: 2022/01/27 14:54 by pzhou

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