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math105-s22:notes:lecture_5 [2022/01/31 23:32] pzhou [Pugh 6.2: construction of measure] |
math105-s22:notes:lecture_5 [2022/02/01 23:35] (current) pzhou |
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====== Lecture 5 ====== | ====== Lecture 5 ====== | ||
$\gdef\mcal{\mathcal{M}}$ | $\gdef\mcal{\mathcal{M}}$ | ||
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Welcome back to in-person instruction. I will continue type in here as a way to prepare for class. | Welcome back to in-person instruction. I will continue type in here as a way to prepare for class. | ||
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- | One statement worth emphasizing is that, " | + | One statement worth emphasizing is that, " |
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+ | Hyperplanes $\{a\}\times \R^{n-1} \In \R^n$ is a null-set. For example, for any $\epsilon> | ||
+ | $$ \{0\} \times \R = \cup_{n=1}^\infty (-\epsilon 2^{-2n-2}, \epsilon 2^{-2n-2}) \times (-2^n, 2^n) $$ | ||
+ | where the sum of area of boxes is less than $\epsilon$. | ||
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+ | ===== Pugh 6.4: Regularity ===== | ||
+ | Our goal here is to prove that, any Lebesgue measurable set is a Borel set plus or minus a null-set. More precisely. $E$ is Lebesgue measurable, if and only if there is a $G_\delta$-set (countable intersection of open) $G$, and an $F_\sigma$-set, | ||
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