LEC ANA1 26

Definition of the integral

Definition 1(Definition).

A partition $P$ of $[a,b]$ is a finite (multi)set of points $x_0,x_1,\dots ,x_n$ where

$$a=x_0\le x_1\le \cdots \le x_n\le b.$$

We write $\Delta x_i\equiv x_i-x_{i-1}$.

Note that a partition has finitely many points.

Riemann integrals

Definition 2(Rudin 6.1).

Suppose $f:[a,b]\to \mathbb{R}$ is bounded. Corresponding to each partition $P$ of $[a,b]$, define

$$\begin{array}{cc}{M}_i\equiv \sup f(x)& x\in [x_{i-1},x_i],\\ m_i\equiv \inf f(x)& x\in [x_{i-1},x_i],\end{array}$$ $$U(P,f)\equiv \sum _{i=1}^n{M}_i\Delta x_i,$$ $$L(P,f)\equiv \sum _{i=1}^n{m}_i\Delta x_i.$$

$U(P,f)$ is called the upper Riemann sum, and $L(P,f)$ is called the lower Riemann sum. Finally, define

$$\text{the upper Riemann integral, }\overline{{\int }_a^b}fdx\equiv \inf U(P,f),\text{ and }$$ $$\text{the lower Riemann integral, }\underline{{\int }_a^b}fdx\equiv \sup L(P,f),$$

where the $\inf$ and $\sup$ are taken over all partitions $P$ of $[a,b]$.

Observe that $L(P,f)\le U(P,f)$ for a given partition $P$, since $m_i\le M_i$. Further, if $M=\sup f(x)$ and $m=\inf f(x)$, $x\in [a,b]$, then

$$m(b-a)\le L(P,f)\le U(P,f)\le M(b-a).$$

Since $L(P,f)$ is bounded above and $U(P,f)$ is bounded below, we can be sure that their supremum and infimum exists respectively, i.e, the upper and lower integrals are defined for every bounded function $f$. Note the use of the LUB property of $\mathbb{R}$.

Definition 3(Definition).

$f$ is Riemann integrable if the upper and lower integrals coincide. We denote this by $f\in \mathcal{R}$. If $f$ is Riemann integrable, we define

$${\int }_a^{b}fdx\equiv \overline{{\int }_a^b}fdx=\underline{{\int }_a^b}fdx.$$

Important

Since the definition of the Riemann integral requires $f$ to be bounded, the statement $f\in \mathcal{R}$ presumes that $f$ is bounded. Ditto for the R-S integral.

Riemann-Stieltjes integrals

Suppose $f:[a,b]\to \mathbb{R}$ is bounded. Let $\alpha$ be a monotone increasing function on $[a,b]$. For a partition $P$ of $[a,b]$, define $\Delta {\alpha }_i\equiv \alpha (x_i)-\alpha (x_{i-1})$. Define

$$U(P,f,\alpha )\equiv \sum _{i=1}^n{M}_i\Delta {\alpha }_i,$$ $$L(P,f,\alpha )\equiv \sum _{i=1}^n{m}_i\Delta {\alpha }_i.$$

The definitions of $M_i$ and $m_i$ are unchanged, and thus are not impacted by $\alpha$. Observe that

$$m(\alpha (b)-\alpha (a))\le L(P,f,\alpha )\le U(P,f,\alpha )\le M(\alpha (b)-\alpha (a)).$$

Now define

$$\overline{{\int }_a^b}fd\alpha \equiv \inf U(P,f,\alpha ),$$ $$\underline{{\int }_a^b}fd\alpha \equiv \sup L(P,f,\alpha ).$$

The $\inf$ and $\sup$ are taken over all partitions, with $\alpha$ being fixed. If the lower and upper Stieltjes integrals are equal, their common value is denoted by

$${\int }_a^{b}fd\alpha ,$$

called the Stieltjes integral of $f$ with respect to $\alpha$, and we say $f$ is R-S or S integrable with respect to $\alpha$, denoted as $f\in \mathcal{R}(\alpha )$. Remember that $d\alpha$ here is just notation.

Observe that

• taking $\alpha (x)=x$ gives us the vanilla Riemann integral.
• $f$ begin Riemann-Stieltjes integrable for some $\alpha$ does not mean $f$ is Riemann-Stieltjes integrable for all $\alpha$.

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Lower integrals, upper integrals and integrability

Definition 4(Definition Rudin, 6.3).

A partition $P^{\ast }$ is called a refinement of $P$ if $P^{\ast }\supset P$. Given two partitions $P_1$ and $P_2$, we say $P^{\ast }$ is their common refinement if $P^{\ast }=P_1\cup P_2$.

Effect of refinement on upper and lower sums

Theorem 5(Rudin 6.4).

$P^{\ast }$ is a refinement of $P$ $⟹$ $L(P,f,\alpha )\le L(P^{\ast },f,\alpha )$ and $U(P,f,\alpha )\ge U(P^{\ast },f,\alpha )$.

Proof.

It is enough to consider the case where$P^{\ast }$ contains a single extra point $x^{\ast }$ with say $x_{i-1}<x^{\ast }<x_i$. Let

$$\begin{array}{cc}{w}_1\equiv \inf f(x)& x\in [x_{i-1},x^{\ast }],\\ w_2\equiv \inf f(x)& x\in [x^{\ast },x_i].\end{array}$$

Note that $w_1,w_2\ge m_i$. Now,

$$\begin{array}{rl}L(P^{\ast },f,\alpha )-L(P,f,\alpha )=& w_1[\alpha (x^{\ast })-\alpha (x_{i-1})]+w_2[\alpha (x_i)-\alpha (x^{\ast })]\\ & -m_i[\alpha (x_i)-\alpha (x_{i-1})]\\ =& (w_1-m_i)[-\alpha (x_{i-1})]+(w_2-m_i)[\alpha (x_i)]\\ & +w_1\alpha (x^{\ast })-w_2\alpha (x^{\ast })-m_i\alpha (x^{\ast })+m_i\alpha (x^{\ast })\\ =& (w_1-m_i)[\alpha (x^{\ast })-\alpha (x_{i-1})]+(w_2-m_i)[\alpha (x_i)-\alpha (x^{\ast })]\\ \ge & 0\end{array}$$ □

Relation between upper and lower integrals

Theorem 6(Rudin 6.5).

$$\underline{{\int }_a^b}fd\alpha \le \overline{{\int }_a^b}fd\alpha .$$

Proof.

Consider arbitrary partitions$P_1$ and $P_2$ of $[a,b]$. We will show that $L(P_1,f,\alpha )\le U(P_2,f,\alpha )$. Let $P^{\ast }$ be the common refinement $P_1\cup P_2$. Then, from the previous theorem, we have

$$L(P_1,f,\alpha )\le L(P^{\ast },f,\alpha )\le U(P^{\ast },f,\alpha )\le U(P_2,f,\alpha ).$$

Now, fix $P_2$. We get $L(P,f,\alpha )\le U(P_2,f,\alpha )$ for all $P$. Thus, $\sup L(P,f,\alpha )\le U(P_2,f,\alpha )$. However, this is true for all $P_2$. Thus, $\sup L(P,f,\alpha )\le \inf U(P,f,\alpha )$.□

A criterion for integrability

Theorem 7(Rudin, 6.6).

$f:[a,b]\to \mathbb{R}\in \mathcal{R}(\alpha )$ $⟺$ $\forall ϵ>0$ $\exists$ partition $P$ of $[a,b]$ such that $U(P,f,\alpha )-L(P,f,\alpha )<ϵ$.

Proof.

($⟸$) If $U(P,f,\alpha )-L(P,f,\alpha )<ϵ$ for some $P$, then $\inf U(P,f,\alpha )-\sup L(P,f,\alpha )<ϵ$. Using 6.5, we have

$$0\le \overline{{\int }_a^b}fd\alpha -\underline{{\int }_a^b}fd\alpha \le ϵ$$

for all $ϵ$. Thus, we must have equality.

($⟹$) Given $ϵ$, find $P_1$ such that $\int fd\alpha -L(P_1,f,\alpha )<\frac{ϵ}{2}$ and $P_2$ such that $U(P_2,f,\alpha )-\int fd\alpha <\frac{ϵ}{2}$. Now, consider the common refinement $P^{\ast }$ of $P_1$ and $P_2$. From 6.4, we get $U(P^{\ast },f,\alpha )-L(P^{\ast },f,\alpha )<ϵ$.□