Real Analysis/Riemann integration

Definition
Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions. The Riemann integral was developed by Bernhard Riemann in 1854 and was, when invented, the first rigorous definition of integration applicable to not necessarily continuous functions.

We will first define some preliminary ideas.

Definition
Let $$a,b\in\R$$

A Partition $$\mathcal{P}$$ is defined as the ordered $$n$$-tuple of real numbers $$\mathcal{P}=(x_0,x_1,\ldots,x_n)$$ such that $$a=x_0<x_1<\ldots<x_n=b$$

Norm of a Partition
Let $$\mathcal{P}$$ be a partition given by $$\mathcal{P}=(x_0,x_1,x_2,\ldots,x_n)$$

Then, the Norm (or the "mesh") of $$\mathcal{P}$$ is defined as $$\|\mathcal{P}\|=\sup\Big\{\big|x_k-x_{k-1}\big|:1\le k\le n\Big\}$$

Tagged Partition
Let $$\mathcal{P}=(x_0,x_1,\ldots,x_n)$$ be a partition

A Tagged Partition $$\mathcal{\dot{P}}$$ is defined as the set of ordered pairs $$\mathcal{\dot{P}}=\Big\{\big([x_{k-1},x_k],t_k\big)\Big\}_{k=1}^n$$ such that $$x_{k-1}\le t_k\le x_k$$. The points $$t_k$$ are called Tags.

Riemann Sums
Let $$f:[a,b]\to\mathbb{R}$$

Let $$\mathcal{\dot{P}}=\Big\{\big([x_{k-1},x_k],t_k\big)\Big\}_{k=1}^n$$ be a tagged partition of $$[a,b]$$

The Riemann Sum of $$f$$ over $$[a,b]$$ with respect to $$\mathcal{\dot{P}}$$ is given by

$$S(f,\mathcal{\dot{P}})=\sum_{k=1}^n f(t_k)(x_k-x_{k-1})$$

Riemann Integral
Let $$f:[a,b]\to\R$$

Let $$L\in\R$$

We say that $$f$$ is Integrable on $$[a,b]$$ if and only if, for every $$\varepsilon>0$$ there exists $$\delta>0$$ such that for every partition $$\mathcal{\dot{P}}$$ satisfying $$\|\mathcal{\dot{P}}\|<\delta$$, we have that $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$

$$L$$ is said to be the integral of $$f$$ over $$[a,b]$$, and is written as

$$L=\int\limits_a^b f(x)dx$$ or as $$L=\int\limits_a^b f$$

Theorem (Uniqueness)
Let $$f:[a,b]\to\R$$ be integrable on $$[a,b]$$

Then the integral $$L$$ of $$f$$ is unique

Proof
Assume, if possible that $$L_1\ne L_2$$ are both integrals of $$f$$ over $$[a,b]$$. Consider $$\varepsilon=\tfrac{|L_1-L_2|}{2}$$

As $$L_1,L_2$$ are integrals, there exist $$\delta_1,\delta_2>0$$ such that $$|S(f,\mathcal{\dot{P}})-L_1|<\varepsilon$$ for all $$\mathcal{\dot{P}}$$ that satisfy $$\|\mathcal{\dot{P}}\|<\delta_1$$ and $$|S(f,\mathcal{\dot{P}})-L_2|<\varepsilon$$ for all $$\mathcal{\dot{P}}$$ that satisfy $$\|\mathcal{\dot{P}}\|<\delta_2$$

Let $$\delta=\inf\{\delta_1,\delta_2\}$$. Hence, if $$\mathcal{\dot{P}}$$ is a partition satisfying $$\|\mathcal{\dot{P}}\|<\delta$$, then we have $$|S(f,\mathcal{\dot{P}})-L_1|<\varepsilon$$ and that $$|S(f,\mathcal{\dot{P}})-L_2|<\varepsilon$$

That is, $$|L_1-L_2|<2\varepsilon=|L_1-L_2|$$, which is an obvious contradiction. Hence the integral $$L$$ of $$f$$ is unique.

We now state (without proof) two seemingly obvious properties of the integral.

Theorem
Let $$f,g:[a,b]\to\R$$ be integrable and let $$c\in(a,b)$$

Then:

(i)$$\int\limits_a^b f+\int\limits_a^b g=\int\limits_a^b(f+g)$$

(ii)$$\int\limits_a^c f+\int\limits_c^b f=\int\limits_a^b f$$

Theorem (Boundedness Theorem)
Let $$f:[a,b]\to\R$$ be Riemann integrable. Then $$f$$ is bounded over $$[a,b]$$

Proof
Assume if possible that $$f$$ is unbounded. For every $$n\in\N$$ divide the interval $$[a,b]$$ into $$n$$ parts. Hence, for every $$n\in\N$$, $$f$$ is unbounded on at least one of these $$n$$ parts. Call it $$I_n$$.

Now, let $$\varepsilon>0$$ be given. Consider an arbitrary $$\delta>0$$. Let $$\mathcal{\dot{P}}$$ be a tagged partition such that $$\|\mathcal{\dot{P}}\|<\delta$$ and $$(I_n,t_n)\in\mathcal{\dot{P}}$$, where $$t_n$$ is taken so as to satisfy $$|f(t_n)|>n\varepsilon$$.

Thus we have that $$|S(f,\mathcal{\dot{P}})-L|>\varepsilon$$. But as $$\delta>0$$ is arbitrary, we have a contradiction to the fact that $$f$$ is Riemann integrable.

Hence, $$f$$ is bounded.

Integrability
We now study classes of Riemann integrable functions. The first "constraint" on Riemann integrable functions is provided by the Cauchy Integrability Criterion.

Theorem (Cauchy Criterion)
Let $$f:[a,b]\to\mathbb{R}$$

Then,

(i)$$f$$ is Riemann integrable on $$[a,b]$$ if and only if

(ii) For every $$\varepsilon>0$$, there exists $$\delta>0$$ such that if $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ are two partitions satisfying $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta$$ then $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\varepsilon$$

Proof
($$\Rightarrow$$)Let $$\int_a^b f=L$$ and let $$\varepsilon>0$$ be given.

Then, there exists $$\delta>0$$ such that for every partition $$\mathcal{\dot{P}}$$ satisfying $$\|\mathcal{\dot{P}}\|<\delta$$,we have $$|S(f,\mathcal{\dot{P}})-L|<\tfrac{\varepsilon}{2}$$

Now, let partitions $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ be such that $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta$$.

Thus we have that $$|S(f,\mathcal{\dot{P}})-L|,|S(f,\mathcal{\dot{Q}})-L|<\tfrac{\varepsilon}{2}$$, that is $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\varepsilon$$

($$\Leftarrow$$) For every $$n\in\mathbb{N}$$, consider $$\delta_n>0$$ such that for all partitions $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ satisfying $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta_n$$, we have $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\tfrac{1}{n}$$.

Without loss of generality, we can assume that $$\delta_m>\delta_n$$ when $$m0$$, we have a $$\delta>0$$ such that $$\|\mathcal{\dot{P}}\|<\delta$$ implies $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$.

Thus $$\int_a^b f=L$$

Theorem (Squeeze Theorem)
Let $$f:[a,b]\to\mathbb{R}$$

Then,

(i) $$f$$ is Riemann integrable on $$[a,b]$$ if and only if

(ii) For every $$\varepsilon>0$$, there exist Riemann integrable functions $$\alpha_{\varepsilon},\omega_{\varepsilon}:[a,b]\to\mathbb{R}$$ such that

$$\alpha_{\varepsilon}(x)\leq f(x)\leq\omega_{\varepsilon}(x)$$ for all $$x\in[a,b]$$ and

$$\int_a^b (\omega_{\varepsilon}-\alpha_{\varepsilon})<\varepsilon$$

Proof
($$\Rightarrow$$)Take $$\alpha_{\varepsilon}(x)=f(x)=\omega_{\varepsilon}(x)$$. It is easy to see that $$\int_a^b (\omega_{\varepsilon}-\alpha_{\varepsilon})<\varepsilon$$

($$\Leftarrow$$)Let $$n\in\mathbb{N}$$. Then, there exist functions $$\alpha_n,\omega_n$$ such that $$\int_a^b (\omega_n-\alpha_n)<\tfrac{1}{n}$$. Further, if $$\int_a^b \alpha_n=A_n$$ and $$\int_a^b \omega_n=Z_n$$, then there exist $$\delta_1,\delta_2>0$$ such that if a partition $$\mathcal{\dot{P}}$$ satisfies $$\|\mathcal{\dot{P}}\|<\delta_1$$ then $$|S(\alpha_n,\mathcal{\dot{P}})-A_n|<\tfrac{1}{n}$$ and $$\|\mathcal{\dot{P}}\|<\delta_2$$ then $$|S(\omega_n,\mathcal{\dot{P}})-Z_n|<\tfrac{1}{n}$$

Now let $$\mathcal{\dot{P}}_n$$ be a partition satisfying $$\|\mathcal{\dot{P}}_n\|<\inf \{\delta_1,\delta_2\}$$.

Now, we can easily see that $$|S(f,\mathcal{\dot{P}}_n)-S(f,\mathcal{\dot{P}}_{n-1})|<\tfrac{1}{n}$$. Hence, $$S(f,\mathcal{\dot{P}}_n)$$ is a Cauchy sequence, with a limit $$L\in\mathbb{R}$$, and as in the previous proof, we can show that $$\int_a^b f=L$$