update

pages/Math4121/Math4121_L17.md

@@ -54,7 +54,7 @@ What is the structure of $S=[0,1]\setminus T$? (or Sparse)
 - Topologically (not dense)
 - Measure, for now meaning small or zero outer content.
 
-## Chapter 4: Nowhere Dense SEts and the Problem with the Fundamental Theorem of Calculus
+## Chapter 4: Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus
 
 ### Nowhere Dense Sets
 

pages/Math4121/Math4121_L18.md

@@ -1 +1,72 @@
-# Lecture 18
+# Math4121 Lecture 18
+
+## Continue
+
+### Small sets
+
+A set that is nowhere dense, has zero outer content yet is uncountable.
+
+By modifying this example, we can find similar with any outer content between 0 and 1.
+
+#### Definition: Perfect Set
+
+$S\subsetes[0,1]$ is perfect if $S=S'$.
+
+Example:
+
+- $[0,1]$ is perfect
+  - perfect sets are closed
+- Finite collection of points is not perfect because they do not have limit points.
+  - perfect sets are uncountable (no countable sets can be perfect)
+
+#### Middle third Cantor set
+
+We construct the set by removing the middle third of the interval.
+
+Let $C_0=[0,1]$, $C_1=[0,\frac{1}{3}]\cup[\frac{2}{3}]$ ...
+
+Continuing this process indefinitely, we define the Cantor set as
+
+$$
+C=\Bigcap_{n=0}^{\infty}C_n
+$$
+
+1. $C_n\subseteq C_{n-1}$
+2. $\ell(C_n)=\ell(C_{n-1})$
+3. Each $C_n$ is closed.
+
+> The algebraic expression for $C_n$, where $a\in[0,1]$, we write as a decimal expansion in base $3$.
+>
+> $$ a=\sum_{n=1}^{\infty} \frac{a_n}{3^n}$$, where $a_n\in\{0,1,2\}$.
+>
+> In this case, $C_0\to C_1$ means deleting all numbers with $a_1=1$. (the same as deleting the interval $[\frac{1}{3},\frac{2}{3}]$)
+>
+> $C_1\to C_2$ means deleting all the numbers with $a_2=1$.$
+>
+> So we can write the set as $$C=\left\{\sum_{n=1}^{\infty}\frac{a_n}{3^n},a_n\in\{0,2\}\right\}$$
+
+#### Proposition 4.1
+
+$C$ is perfect and nowhere dense, and outer content is 0.
+
+Proof:
+
+(i) $c_e(C)=0$
+
+Let $\epsilon>0$, then $\exists n$ such that $\left(\frac{2}{3}\right)<\epsilon$. Then $C_n$ is a cover of $C$, and $\ell(C_n)<\epsilon$.
+
+(ii) $C$ is perfect
+
+Since $C_n$ is closed, $C$ is closed (any intersection of closed set is closed) so $C'\subseteq C$.
+
+Let $a\in C$, and we need to show $a$ is a limit point. Let $\epsilon>0$, and we need to find $a^*\in C\setminus\{a\}$ and $|a^* - a| < \epsilon$. Suppose $a=\sum_{n=1}^{\infty} \frac{a_n}{3^n}, a_n \in \{0, 2\}$, Notive that if $a^*\in C$ has the expansion as $a$ except the k-th term.
+
+So $|a^*-a|=\frac{2}{3^k}$, which can be made arbitrarily small by choosing a sufficiently large $k$. Thus, $a$ is a limit point of $C$, proving that $C$ is perfect.
+
+(iii) $C$ is nowhere dense
+
+It is sufficient to show $C$ contains no intervals.
+
+Any open intervals has a real number with 1 in it's base 3 decimal expansion (proof in homework)
+
+_take some interval in $(a,b)$ we can change the digits that is small enough and keep the element still in the set_

pages/Math4121/_meta.js

@@ -20,9 +20,7 @@ export default {
     Math4121_L15: "Introduction to Lebesgue Integration (Lecture 15)",
     Math4121_L16: "Introduction to Lebesgue Integration (Lecture 16)",
     Math4121_L17: "Introduction to Lebesgue Integration (Lecture 17)",
-    Math4121_L18: {
-        display: 'hidden'
-    },
+    Math4121_L18: "Introduction to Lebesgue Integration (Lecture 18)",
     Math4121_L19: {
         display: 'hidden'
     },