format updates

content/Math4121/Math4121_L28.md

@@ -31,7 +31,8 @@ $$
 > $$m_e\left(S\cap \bigcup_{j=1}^{\infty} I_j\right) = \sum_{j=1}^{\infty} m_e(S\cap I_j)$$
 > Proved on Friday
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 $\implies$ If Lebesgue criterion holds for $S$, then for any $X$ of finite outer measure,
 
@@ -72,7 +73,7 @@ m_e(X)&\leq m_e(X\cap S)+m_e(S^c\cap X)\\
 \end{aligned}
 $$
 
-QED
+</details>
 
 ### Revisit Borel's criterion
 
@@ -88,7 +89,8 @@ $$
 m_e(S)=\sum_{j=1}^{\infty} m_e(S_j)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 First we prove $m_e(\bigcup_{j=1}^{\infty} S_j)=\sum_{j=1}^{\infty} m(S_j)$ by induction.
 
@@ -116,16 +118,17 @@ Therefore, $\sum_{j=1}^{\infty} m(S_j)\leq m_e(S)\leq \sum_{j=1}^{\infty} m(S_j)
 
 So $S$ is measurable.
 
-QED
+</details>
 
 #### Proposition 5.9 (Preview)
 
 Any finite union (and intersection) of measurable sets is measurable.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Let $S_1, S_2$ be measurable sets.
 
 We prove by verifying the Caratheodory's criteria for $S_1\cup S_2$.
 
-QED
+</details>