This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
math105-s22:notes:lecture_6 [2022/02/02 23:45] pzhou |
math105-s22:notes:lecture_6 [2022/02/03 16:28] (current) pzhou [Lecture 6] |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== Lecture 6 ====== | ====== Lecture 6 ====== | ||
+ | [[https:// | ||
===== 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 $m(E) \times m(F) = m(E \times F)$. | ||
Line 35: | Line 35: | ||
* 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? | ||