update

pages/Math4121/Math4121_L37.md

@@ -41,7 +41,7 @@ For the general case,
 By the Monotone Convergence Theorem (use $|f|\chi_{[-N,N]}$ to approximate $|f|$), we can find $N$ large such that
 
 $$
-\int_{E_N^c}|f|dm<\frac{\epsilon}{2}
+\int_{E_N^c}|f|dm<\frac{\epsilon}{3}
 $$
 
 where $E_N=E\cap [-N,N]$.
@@ -54,8 +54,72 @@ $$
 \int_{E_N} |f-\phi|dm<\frac{\epsilon}{3}
 $$
 
+For each $i=1,2,\cdots,n$, we can find $g_i$ continuous such that
 
+$$
+\int_{E}|\chi_{S_i}-g_i|dm<\frac{\epsilon}{3M}
+$$
 
+where $M=\sum_{i=1}^n |\alpha_i|$.
+
+Take $g=\sum_{i=1}^n \alpha_i g_i$,
+
+$$
+\int_E |\phi-g|dm\leq \sum_{i=1}^n |\alpha_i|\int_E |g_i-\chi_{S_i}|dm<\frac{\epsilon}{3}
+$$
+
+$\phi-g=\sum_{i=1}^n \alpha_i (\chi_{S_i-g_i})$
+
+All in all,
+
+$$
+\begin{aligned}
+\int_E |f-g|dm&\leq \int_E|f-\phi|dm+\int_E |\phi-g|dm\\
+&=\int_{E_N^c}|f|dm+\int_E |f-\phi|dm+\int_E |\phi-g|dm\\
+&<\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}\\
+&=\epsilon
+\end{aligned}
+$$
 
 QED
 
+### Road map for proving the fundamental theorem of calculus in Lebesgue integration
+
+Recall the Riemann-Stieltjes integral:
+
+If $g\in \mathscr{R}(\alpha)$ on $[a,b]$,
+
+$G(x)=\int_a^x g d\alpha$,
+
+then:
+
+1. $G$ is continuous on $[a,b]$
+2. If $g$ is continuous at $x\in [a,b]$, then $G$ is differentiable at $x$ with $G'(x)=g(x)$.
+
+To extend this to the case where $g$ is Lebesgue integrable, we use the Hardy-Littlewood maximal function.
+
+#### Definition of the Hardy-Littlewood maximal function
+
+Given an interval $I\subseteq \mathbb{R}$, define the averaging operator $A_I f(x)=\frac{\chi_I(x)}{m(I)}\int_I f(x)dm$.
+
+(This function takes the average of $f$ over the interval $I$.)
+
+The Hardy-Littlewood maximal function is defined as:
+
+$$
+f^*(x)=\sup_{I\text{ is open interval}}A_I f(x)
+$$
+
+We will show that $f^*$ is not that such worse than $f$. (Prove on Wednesday)
+
+Relates to the Fundamental Theorem of Calculus in Lebesgue integration.
+
+$$
+\frac{G(x+h)-G(x)}{h}=\frac{1}{h}\int_x^{x+h} g(t)dt=A_{[x,x+h]}g(x)
+$$
+
+If we can control all the averages, we can control the function.
+
+
+
+

pages/Math4121/_meta.js

@@ -39,4 +39,8 @@ export default {
     Math4121_L33: "Introduction to Lebesgue Integration (Lecture 33)",
     Math4121_L34: "Introduction to Lebesgue Integration (Lecture 34)",
     Math4121_L35: "Introduction to Lebesgue Integration (Lecture 35)",
+    Math4121_L36: "Introduction to Lebesgue Integration (Lecture 36)",
+    Math4121_L37: "Introduction to Lebesgue Integration (Lecture 37)",
+    Math4121_L38: "Introduction to Lebesgue Integration (Lecture 38)",
+    Math4121_L39: "Introduction to Lebesgue Integration (Lecture 39)",
 }

pages/Math416/Math416_L26.md

@@ -0,0 +1 @@
+

pages/Math416/Math416_L27.md

@@ -0,0 +1 @@
+

pages/Math416/_meta.js

@@ -29,4 +29,6 @@ export default {
     Math416_L23: "Complex Variables (Lecture 23)",
     Math416_L24: "Complex Variables (Lecture 24)",
     Math416_L25: "Complex Variables (Lecture 25)",
+    Math416_L26: "Complex Variables (Lecture 26)",
+    Math416_L27: "Complex Variables (Lecture 27)",
 }