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--- 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 ======
+[[https://berkeley.zoom.us/rec/share/E9YcGb4kx_xuNLrBAVWTWhkZt0Y70DbhrtUMeBjTk6gZYhA8393qWN6W5UO9Yl0_.bMFQSrS0aqiGTY6X| video]]
 ===== Theorem 21 =====
 If  $E \In \R^n, F \In \R^k$  are measurable, then  $E \times F$  is measurable, with  $m(E) \times m(F) = m(E \times F)$ .
@@ 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<i} U(x_j)) $,$ V_i = V(x_i) $, then we still have$ U_i \times V_i \supset \pi^{-1} U_i $. Thus,$ U_i $are disjoint, and we have$ m (\cup U_i) \leq 1 $and$ m (K) \leq \sum_i m(U_i)\times m(V_i) \leq \epsilon$.
+===== 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?

Lecture Notes