format updates

This commit is contained in:
Zheyuan Wu
2025-09-24 01:27:46 -05:00
parent e59ef423f3
commit 143d77e7f9
16 changed files with 401 additions and 79 deletions

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@@ -14,7 +14,8 @@ $$
where $\partial S$ is the boundary of $S$ and $c_e(\partial S)=0$.
Example:
<details>
<summary>Examples for Jordan measurable</summary>
1. $S=\mathbb{Q}\cap [0,1]$ is not Jordan measurable.
@@ -56,6 +57,8 @@ So $c_e(SVC(4))=\frac{1}{2}$.
> General formula for $c_e(SVC(n))=\frac{n-3}{n-2}$, and since $SVC(n)$ is nowhere dense, $c_i(SVC(n))=0$.
</details>
### Additivity of Content
Recall that outer content is sub-additive. Let $S,T\subseteq \mathbb{R}^n$ be disjoint.
@@ -80,7 +83,8 @@ $$
c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i)
$$
Proof:
<details>
<summary>Proof</summary>
$$
\begin{aligned}
@@ -96,7 +100,7 @@ $$
c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i)
$$
QED
</details>
##### Failure for countable additivity for Jordan content