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math105-s22:hw:hw2 [2022/01/27 14:36] pzhou created |
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. | ||
- | How did The North Face get its name? The name of the company is based on the north face of the Half Dome in Yosemite, California, to which attention was given on the generalization that the north face of a mountain in the northern hemisphere is regarded as the coldest, iciest and thus the most formidable to climb. | ||
- | So far, we have climbed the 'one face' of the mountain Lebesgue measure, let's climb the other face. Here are some guides. Our goal is to start from the alternative definition of measurability (see [[math105-s22: | + | So far, we have climbed the 'one face' of the mountain Lebesgue measure, let's climb the other face. ((How did The North Face get its name? The name of the company is based on the north face of the Half Dome in Yosemite, California, to which attention was given on the generalization that the north face of a mountain in the northern hemisphere is regarded as the coldest, iciest and thus the most formidable to climb.)) |
**Def 2 **: A subset is measurable, if for any $\epsilon> | **Def 2 **: A subset is measurable, if for any $\epsilon> | ||
- | and derive the same set of properties for measurable sets. Note that, all the properties of outer-measure can still be used. I am going to be lazy, and steal the proposition we need to prove from Tao's graduate level measure theory book. | + | and derive the same set of properties for measurable sets. Note that, all the properties of outer-measure can still be used. |
+ | In the following, the measurability is defined using Def 2 above. | ||
+ | ==== Lemma 0 ==== | ||
+ | (stolen from Tao's grad [[https:// | ||
- | ==== Lemma 1 ==== | ||
{{: | {{: | ||
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+ | ==== Lemma 1 ==== | ||
+ | Let be any subset of , then | ||
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+ | |||
+ | ==== Lemma 2 ==== | ||
+ | If is a countable collection of measurable set, then is measurable. | ||
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+ | ==== Lemma 3 ==== | ||
+ | Every closed subset is measurable. | ||
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+ | Hint: This is a hard one. First prove that can be written as countable union of bounded closed subsets, then suffice to prove the claim that any bounded closed (hence compact) subset is measurable. | ||
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+ | Hint 2: Given a closed bounded subset of , for any $\epsilon> | ||
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+ | What does a closed set look like? Say, the cantor set in $[0, | ||
+ | |||
+ | ==== Lemma 4 ==== | ||
+ | If is measurable, then is measurable. | ||
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+ | Hint: Try to write as a countable union of closed sets, union a set of measure zero, hence is countable. | ||
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