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-# Lecture 6
\ No newline at end of file
+# Lecture 6
+
+## Chapter 6: Riemann-Stieltjes Integral
+
+> A nice point to restart your learning, LOL.
+
+### Differentiation and existence of the integral
+
+#### Definition 6.1
+
+Let $[a,b]\subseteq \mathbb{R}$. A partition of $[a,b]$ is a finite sequence of points $\{x_0,x_1,\cdots,x_n\}\subseteq [a,b]$ such that $x_0<x_1<\cdots<x_n$.
+
+Let $\alpha:[a,b]\to \mathbb{R}$ be monotone increasing. ($\alpha(x)\leq \alpha(y)$ for $x\leq y$)
+
+_We will use $\alpha$ for monotone increasing function in later sections._
+
+#### Definition 6.2
+
+For a partition $P$ of $[a,b]$, we define $\Delta \alpha_i=\alpha(x_i)-\alpha(x_{i-1})$ for $i=1,2,\cdots,n$.
+
+Let $f:[a,b]\to \mathbb{R}$ be bounded.
+
+Then we define
+
+$$
+m_i=\inf_{x\in [x_{i-1},x_i]}f(x),\quad M_i=\sup_{x\in [x_{i-1},x_i]}f(x).
+$$
+
+Defined the lower and upper Riemann sum by
+
+$$
+L(P,f,\alpha)=\sum_{i=1}^n m_i\Delta \alpha_i,\quad U(P,f,\alpha)=\sum_{i=1}^n M_i\Delta \alpha_i.
+$$
+
+Defined the lower and upper integral by
+
+$$
+\underline{\int_a^b}f(x)d\alpha=\sup_P L(P,f,\alpha),\quad \overline{\int_a^b}f(x)d\alpha=\inf_P U(P,f,\alpha).
+$$
+
+If $\underline{\int_a^b}f(x)d\alpha=\overline{\int_a^b}f(x)d\alpha$, then we say $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a,b]$, written as $f\in \mathscr{R}(\alpha)$, and the common value is called the Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $[a,b]$, denoted by
+
+$$
+\int_a^b f(x)d\alpha=\underline{\int_a^b}f(x)d\alpha=\overline{\int_a^b}f(x)d\alpha.
+$$
+
+If $\alpha(x)=x$, then we write $\int_a^b f(x)dx$ instead of $\int_a^b f(x)d\alpha$.
+
+> Damn, that's a really loooong definition.
+
+#### Definition 6.3
+
+A partition $P^*$ is called a **refinement** of $P$ if $P\subseteq P^*$.
+
+Given two partitions $P_1$ and $P_2$, we define their common refinement $P^*=P_1\cup P_2$. _we can merge two partitions by adding all points in both partitions._
+
+#### Theorem 6.4
+
+If $P^*$ is a refinement of $P$, then
+
+$$
+L(P^*,f,\alpha)\geq L(P,f,\alpha)
+$$
+
+_Refinement of a partition will never make the lower sum smaller._
+
+$$
+U(P^*,f,\alpha)\leq U(P,f,\alpha)
+$$
+
+_Refinement of a partition will never make the upper sum larger._
+
+Proof:
+
+Main idea:
+
+Let $P=P_0\subset P_1\subset P_2\subset \cdots \subset P_K=P^*$.
+
+Where $P_k$ has more points than $P_{k-1}$.
+
+It suffices to show that $L(P_k,f,\alpha)\geq L(P_{k-1},f,\alpha)$ for all $k=1,2,\cdots,K$.
+
+Let $P_{k-1}=\{y_0,y_1,\cdots,y_J\}$ and $P_k=\{y_0,y_1,\cdots,y_{j-1},y^*,y_j,\cdots,y_J\}$.
+
+Then, since $\alpha$ is monotone increasing, we have $y_{j-1}\leq y^*\leq y_j$.
+
+$$
+\begin{aligned}
+L(P_k,f,\alpha)-L(P_{k-1},f,\alpha)&=\sum_{i=1}^{j+1}\inf_{x\in [y_{i-1},y_i]}f(x)(\alpha(y_i)-\alpha(y_{i-1}))-\sum_{i=1}^j\inf_{x\in [y_{i-1},y_i]}f(x)(\alpha(y_i)-\alpha(y_{i-1}))\\
+&=\inf_{x\in [y^*,y_j]}f(x)(\alpha(y_j)-\alpha(y^*))+\inf_{x\in [y_{j-1},y^*]}f(x)(\alpha(y^*)-\alpha(y_{j-1}))-\inf_{x\in [y_{j-1},y_j]}f(x)(\alpha(y_j)-\alpha(y_{j-1}))\\
+&\geq m_j(\alpha(y_j)-\alpha(y^*))+m_j(\alpha(y^*)-\alpha(y_{j-1}))-m_{j-1}(\alpha(y_j)-\alpha(y_{j-1}))\\
+&=0
+\end{aligned}
+$$
+
+Same for $U(P_k,f,\alpha)\geq U(P_{k-1},f,\alpha)$.
+
+EOP
+
+#### Theorem 6.5
+
+$$
+\underline{\int_a^b}f(x)d\alpha\leq \overline{\int_a^b}f(x)d\alpha
+$$
+
+Proof:
+
+Let $P^*$ be a common refinement of $P_1$ and $P_2$.
+
+By **Theorem 6.4**, we have
+
+$$
+L(P_1,f,\alpha)\leq L(P^*,f,\alpha)\leq U(P^*,f,\alpha)\leq U(P_2,f,\alpha)
+$$
+
+Fixing $P_1$ and take the supremum over all $P_2$, we have
+
+$$
+\underline{\int_a^b}f(x)d\alpha\leq \sup_{P_1}L(P_1,f,\alpha)\leq \inf_{P_2}U(P_2,f,\alpha)=\overline{\int_a^b}f(x)d\alpha
+$$
+
+EOP