update

pages/Math4121/Math4121_L22.md

@@ -2,8 +2,6 @@
 
 ## Continue on Arzela-Osgood Theorem
 
-
-
 Proof:
 
 Part 2: Control the integral on $\mathcal{U}$
@@ -100,7 +98,7 @@ Part 2: If $f$ is pointwise discontinuous, then $\mathcal{D}$ is of first catego
 
 Let $P_k=\{x\in [a,b]: w(f;x)\geq \frac{1}{k}\}$, $\mathcal{D}=\bigcup_{k=1}^\infty P_k$.
 
-Need to show that each $P_k$ is nowhere dense. (under the assumption that $\mathcal{C)$ is dense).
+Need to show that each $P_k$ is nowhere dense. (under the assumption that $\mathcal{C}$ is dense).
 
 Let $I\subseteq [a,b]$ so $\exists c\in \mathcal{C}\cap I$. So by definition of $w(f;c)$, $\exists J\subseteq I$ and $c\in J$ such that $w(f;J)\leq \frac{1}{k}$ so for all $x\in J$, $w(f;x)\leq \frac{1}{k}$. so $J\subseteq P_k=\emptyset$.
 

pages/Math4121/Math4121_L23.md

@@ -1 +1,139 @@
-# Lecture 23
+# Math 4121 Lecture 23
+
+## Chapter 5 Measure Theory
+
+### Weierstrass idea
+
+Define
+
+$$
+S_f(x) = \{(x,y)\in \mathbb{R}^2: 0\leq y\leq f(x)\}
+$$
+
+We take the outer content in $\mathbb{R}^2$ of $S_f(x)$ to be the area of the largest rectangle that can be inscribed in $S_f(x)$.
+
+$$
+(w)\int_a^b f(x) dx = c_e(S_f(x))
+$$
+
+We can generalize this to higher dimensions.
+
+#### Definition volume of rectangle
+
+Let $R=I_1\times I_2\times \cdots \times I_n\in \mathbb{R}^n$ be a rectangle.
+
+The volume of $R$ is defined as
+
+$$
+\text{vol}(R) = \prod_{i=1}^n \ell(I_i)
+$$
+
+#### Definition of outer content
+
+For $S\subseteq \mathbb{R}^n$, we define the outer content of $S$ as
+
+$$
+c_e(S) = \inf_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
+$$
+
+where $S\subseteq \bigcup_{j=1}^N R_j$ and $R_j$ are rectangles.
+
+Note: $\overline{\int}f(x) dx=c_e(S_f(x))$
+
+#### Definition of inner content
+
+For $S\subseteq \mathbb{R}^n$, we define the inner content of $S$ as
+
+$$
+c_i(S) = \sup_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
+$$
+
+where $R_j$ are disjoint rectangles $\in \mathbb{R}^n$ and $\bigcup_{j=1}^N R_j\subseteq S$.
+
+Note: $\underline{\int}f(x) dx=c_i(S_f(x))$
+
+#### Definition of Jordan measurable set
+
+A set $S\subseteq \mathbb{R}^n$ is said to be _Jordan measurable_ if $c_e(S)=c_i(S)$.
+
+and we denote the common value **content** as $c_e(S)=c_i(S)=c(S)$.
+
+#### Definition of interior of a set
+
+The interior of a set $S\subseteq \mathbb{R}^n$ is defined as
+
+$$
+S^\circ = \{x\in \mathbb{R}^n: B_\delta(x)\subseteq S \text{ for some } \delta > 0\}
+$$
+
+_It is the largest open set contained in $S$._
+
+#### Definition of closure of a set
+
+The closure of a set $S\subseteq \mathbb{R}^n$ is defined as
+
+$$
+\overline{S} = S\cup S'
+$$
+
+or equivalently,
+
+$$
+\overline{S} = \{x\in \mathbb{R}^n: B_\delta(x)\cap S\neq \emptyset \text{ for all } \delta > 0\}
+$$
+
+where $S'$ is the set of all limit points of $S$.
+
+_It is the smallest closed set containing $S$._
+
+Homework problem: Complement of the closure of $S$ is the interior of the complement of $S$, i.e.,
+
+$$
+(\overline{S})^c = (S^c)^\circ
+$$
+
+#### Definition of boundary of a set
+
+The boundary of a set $S\subseteq \mathbb{R}^n$ is defined as
+
+$$
+\partial S = \overline{S}\setminus S^\circ
+$$
+
+#### Proposition 5.1 (Criterion for Jordan measurability)
+
+Let $S\subseteq \mathbb{R}^n$ be a bounded set. Then
+
+$$
+c_e(S) = c_i(S)+c_e(\partial S)
+$$
+
+So $S$ is Jordan measurable if and only if $c_e(\partial S)=0$.
+
+Proof:
+
+Let $\epsilon > 0$, and $\{R_j\}_{j=1}^N$ be an open cover of $\partial S$. such that $\sum_{j=1}^N \text{vol}(R_j) < c_e(\partial S)+\frac{\epsilon}{2}$.
+
+We slightly enlarge each $R_j$ to $Q_j$ such that $R_j\subseteq Q_j$ and $\text{vol}(Q_j)\leq \text{vol}(R_j)+\frac{\epsilon}{2N}$.
+
+and $dis(R_j,Q_j^c)>\delta > 0$
+
+If we could construct such $\{Q_j\}_{j=N+1}^M$ disjoint and
+
+$$
+\bigcup_{j=N+1}^M Q_j\subseteq S\subseteq \bigcup_{j=1}^M Q_j
+$$
+
+then we have
+
+$$
+c_e(S)\leq \sum_{j=1}^M \text{vol}(\partial S)+\epsilon +c_i(S)
+$$
+
+We can do this by constructing a set of square with side length $\eta$. We claim:
+
+If $\eta$ is small enough (depends on $\delta$), then $\mathcal{C}_\eta=\{Q\in K_\eta:Q\subset S\}$, $\mathcal{C}_\eta\cup \left(\bigcup_{j=1}^N Q_j\right)$ is a cover of $S$.
+
+Suppose $\exists x\in S$ but not in $\mathcal{C}_\eta$. Then $x$ is closed to $\partial S$ so in some $Q_j$. (This proof is not rigorous, but you get the idea. Also not clear in book actually.)
+
+EOP