math105-s22:notes:lecture_6
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math105-s22:notes:lecture_6 [2022/02/03 07:45] pzhou |
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| ====== Lecture 6 ====== | ====== Lecture 6 ====== | ||
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| ===== Theorem 21 ===== | ===== Theorem 21 ===== | ||
| If $E \In \R^n, F \In \R^k$ are measurable, then $E \times F$ is measurable, with $m(E) \times m(F) = m(E \times F)$. | If $E \In \R^n, F \In \R^k$ are measurable, then $E \times F$ is measurable, with $m(E) \times m(F) = m(E \times F)$. | ||
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| * We know $K \subset \cup_x U(x) \times V(x)$, but that's uncountably many set. We can pass to a finite subcover, indexed by $x_1, \cdots, x_N$. Let $U_i = U(x_i) \RM (\cup_{j< | * We know $K \subset \cup_x U(x) \times V(x)$, but that's uncountably many set. We can pass to a finite subcover, indexed by $x_1, \cdots, x_N$. Let $U_i = U(x_i) \RM (\cup_{j< | ||
| + | ===== Discussion ===== | ||
| + | - Can you prove that $\{y=x\} \In \R^2$ has measure $0$? | ||
| + | - In both of the two proofs above, we assumed $E$ was bounded, how to deal with the general case? | ||
| + | - Prove that every closed subset (e.g. your favorite Cantor set is a closed set) in $\R$ is a $G_\delta$-set. Is it true that every open set is a $F_\sigma$-set? | ||
math105-s22/notes/lecture_6.1643874304.txt.gz · Last modified: 2026/02/21 14:43 (external edit)
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