update

pages/Math4121/Math4121_L12.md

@@ -1,4 +1,4 @@
-# Lecture 12
+# Math4121 Lecture 12
 
 ## Chapter 7: Uniform Convergence and Integrals
 

pages/Math4121/Math4121_L14.md

@@ -2,9 +2,9 @@
 
 ## Recap
 
-### Hankel developedn Riemann's integrabilty criterion.
+### Hankel developed Riemann's integrabilty criterion.
 
-#### Definition 
+#### Definition: Oscillation
 
 Given an interval $I\subset[a,b]$, $f:[a,b]\to\mathbb{R}$ the oscillation of $f$ at $I$ is
 
@@ -12,8 +12,88 @@ $$
 \omega(f,I) = \sup_I f - \inf_I f
 $$
 
-#### Theorem 
+#### Theorem 2.5: Riemann's Integrability Criterion
 
-A bounded function $f$ is Riemann integrable if and only if
+A bounded function $f$ is Riemann integrable if and only if for every $\sigma,\epsilon>0$ there exists a partition $P$ of $[a,b]$ such that
+
+$$
+\sum_{i\in \mathcal{P}}\Delta x_i<\epsilon
+$$
+
+where $\mathcal{P}=\{i:\omega(f,I_i)>\sigma\}$.
+
+Proof as homework questions.
+
+#### Corollary 2.4
+
+For point $c\in[a,b]$, define the oscillation at $c$ as
+
+$$
+\omega(f,c) = \inf_{c\in I}\omega(f,I)
+$$
+
+Homework question 6: $f$ is continuous at $c$ if and only if $\omega(f,c)=0$.
+
+So we can restate the previous theorem as:
+
+Given $\sigma>0$, define $S_\sigma=\{c\in[a,b]:\omega(f,c)>\sigma\}$.
+
+Restate the theorem as:
+
+$f\in\mathscr{R}[a,b]$ if and only if for every $\sigma,\epsilon>0$ there exists intervals $I_1,I_2,\cdots,I_n$ such that $S_\sigma\subset\bigcup_{i=1}^{n}I_i$ and $\sum_{i=1}^{n}\ell(I_i)<\epsilon$. where $\ell(I)$ is the length of the interval $I$.
+
+#### Definition: Outer content
+
+Given a set $S$, a **finite cover** of $S$ is a collection of intervals $C=\{I_1,I_2,\cdots,I_n\}$ such that $S\subseteq\bigcup_{i=1}^{n}I_i$.
+
+The length of the cover $C$ is $\ell(C)=\sum_{i=1}^{n}\ell(I_i)$.
+
+The **outer content** of $S$ is
+
+$$
+c_e(S) = \inf_{c\in C_s}\ell(C)
+$$
+
+where $\C_s$ is the set of all finite covers of $S$.
+
+Example: 
+
+$S=\{x_1,\ldots,x_n\}$, then $c_e(S)=0$. 
+
+- Let $I_i=(x_i-\frac{\epsilon}{2n},x_i+\frac{\epsilon}{2n})$, so $\sum_{i=1}^{n}\ell(I_i)=\epsilon$
+
+$S=\{\frac{1}{n}\}_{n=1}^{\infty}$, then $c_e(S)=0$.
+
+- In this case, we can only use finite cover, however, there is only one "accumulation point", so we can cover it with a single interval, and the remaining points can be covered by finite intervals. (for any $\epsilon>0$, we can construct a **finite cover** with length $\epsilon$ that covers all points.)
+
+$S=\mathbb{Q}\cap[0,1]$, then $c_e(S)=1$.
+
+- In this case, by covering the interval with $[0,1]$, we can get the length of the cover is at most 1.
+- Suppose there exists a cover $C$ with $\sum_{I\in C}\ell(I)<1$, then there must be a gap in the intervals, however, since the $\mathbb{Q}$ is dense in $\mathbb{R}$, there must be a point in the gap, which is a contradiction.
+
+#### Theorem 2.5: Hankel's criterion for Riemann integrability
+
+A function $f\in\mathscr{R}[a,b]$ if and only if $c_e(S_\sigma)=0$ for all $\sigma>0$.
+
+_The idea is that if the oscillation of a function can be bounded by a finite set that the total length is small, then the function is Riemann integrable._
+
+Hankel's idea was to apply this theorem to determining how discontinuous a function could be a Riemann integrable function.
+
+> A set $S$ is dense in $X$ if every point of $X$ is a limit point of $S$.
+
+#### Definition: Totally discontinuous
+
+$f$ is **totally discontinuous** if the points of continuity of $f$ are not dense.
+
+For example, $f(x)=\begin{cases}
+0 & x\in\mathbb{Q}\\
+1 & x\notin\mathbb{Q}
+\end{cases}$ is totally discontinuous.
+
+#### Definition: Pointwise discontinuity
+
+$f$ is **pointwise discontinuous** if they are dense in $[a,b]$.
+
+Hankel's conjecture: $f$ is pointwise discontinuous, then $f$ is integrable.
 
 

pages/Math4121/index.md

@@ -1,6 +1,6 @@
 # Math 4121
 
-Riemann integration; measurable functions; measures; the Lebesgue integral; integrable functions; $L^p$ spaces.
+Riemann integration; measurable functions; Measures; the Lebesgue integral; integrable functions; $L^p$ spaces.
 
 ## Textbook