Math4121 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

Defined the lower and upper Riemann sum by

Defined the lower and upper integral by

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

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

Refinement of a partition will never make the lower sum smaller.

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 $y_{j-1}\leq y^*\leq y_j$.

Same for $U(P_k,f,\alpha)\geq U(P_{k-1},f,\alpha)$.

QED

Theorem 6.5

Proof:

Let $P^*$ be a common refinement of $P_1$ and $P_2$.

By Theorem 6.4, we have

Fixing $P_1$ and take the supremum over all $P_2$, we have

QED