format updates

content/Math4121/Math4121_L27.md

@@ -14,7 +14,7 @@ where $I_j$ is an open interval
 
 1. $m_e(I)=\ell(I)$
 2. Countably sub-additive: $m_e\left(\bigcup_{n=1}^\infty S_n\right)\leq \sum_{n=1}^\infty m_e(S_n)$ (Prove today)
-3. does not repect complementation (Build in to Borel measure)
+3. does not respect complementation (Build in to Borel measure)
 
 Why does Jordan content respect complementation?
 
@@ -64,7 +64,8 @@ $$
 m_e\left(\bigcup_{n=1}^\infty S_n\right)\leq\sum_{n=1}^\infty m(S_n)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Let $\epsilon>0$ and for each $j$, let $\{I_{i,j}\}_{i=1}^\infty$ be a cover of $S_j$ s.t.
 
@@ -84,21 +85,22 @@ $$
 m_e\left(\bigcup_{j=1}^\infty S_j\right)\leq\sum_{j=1}^\infty m_e(S_j)=\sum_{j=1}^\infty m(S_j)
 $$
 
-QED
+</details>
 
-#### Corollary
+#### Corollary: inner measure is always less than or equal to outer measure
 
 $$
 m_i(S)\leq m_e(S)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 $$
 m_i(S)=m(I)-m_e(I\setminus S)\leq m(I)-m_i(I\setminus S)=m_e(S)
 $$
 
-QED
+</details>
 
 ### Caratheodory's Criterion
 
@@ -110,7 +112,8 @@ $$
 m_e\left(S\cap \left(\bigcup_{j=1}^\infty I_j\right)\right)=m_e\left(\bigcup_{j=1}^\infty (S\cap I_j)\right)=\sum_{j=1}^\infty m_e(S\cap I_j)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 For each $j$, let $\{J_i\}_{i=1}^\infty$ be a cover of $S\cap \left(\bigcup_{j=1}^\infty I_j\right)$ such that $\sum_{i=1}^\infty \ell(J_i)<c_e(S\cap \left(\bigcup_{j=1}^\infty I_j\right))+\epsilon$. Since $\{I_j\}_{j=1}^\infty$ are pairwise disjoint, so is $\{J_i\cap I_j\}_{j=1}^\infty$ for each $i$.
 
@@ -132,7 +135,7 @@ $$
 m_e\left(S\cap \left(\bigcup_{j=1}^\infty I_j\right)\right)\leq \sum_{j=1}^\infty m_e(S\cap I_j)
 $$
 
-QED
+</details>
 
 #### Theorem 5.6 (Caratheodory's Criterion)