Riemann Integral

The Riemann Integral was the first formal definition of an integral, defined by Bernhard Riemann in 1854. Gaston Darboux developed a simpler approach to the Riemann Integral in 1875 (source needed), that I’ll be using in this section.

Partitions

Definition

Let [a,b] be an interval in \mathbb{R}. A partition P of [a,b] is a finite set of points \{x_0,\ldots,x_n\} in [a,b] such that

a = x_0 < x_1 < \cdots < x_n = b.

A partition serves to chop up our interval [a,b] into fintely many subintervals [x_{i-1},x_{i}]. We notate the length of the i^{th} sub-interval in our partition with \Delta x_i = x_i - x_{i-1}.

Definition

If P and Q are two partitions of [a,b] with P \subseteq Q, then Q is called a refinement of P.

Darboux sums

Definition

Let f be a bounded function on [a,b] and P = \{x_0,\ldots,x_n\} be any partition on [a,b]. Let

M_i(f) = \sup\{f(x) \colon x \in [x_{i-1},x_i]\} and m_i(f) = \inf\{f(x) \colon x \in [x_{i-1},x_i]\}

for each i \in \{1,\ldots,n\}. We define U(f,P) = \sum^n_{i=1}M_i\Delta x_i \qquad L(f,P) = \sum^n_{i=1}m_i\Delta x_i

as the Upper sum and Lower sum respectively.

Because we assume f to be bounded function, there exists a lower and upper bound on the interval [a,b]. Call these bounds m,M respectively (No relation to m_i,M_i). For any partition P of [a,b] we have m(b-a) \leq L(f,p) \leq U(f,p) \leq M(b-a).

Thus we know the upper and lower sums for f exist, and form a bounded set.

Theorem 4.1.1

Let f be a bounded function on [a,b]. If P and Q are partitions of [a,b], and Q is a refinement of P, then

L(f,P) \leq L(f,Q) \leq U(f,Q) \leq U(f,P).

Direct Proof

The middle inequality L(f,Q) \leq U(f,Q) above follows from our definitions. To prove L(f,P) \leq L(f,Q), suppose P = \{x_0,\ldots,x_k\}, and consider the partition P* formed by adding x* to P, where x_{k-1} < x* < x_k for some k \in \{1,\ldots,n\}. Let

\begin{align*} t_1 &= \inf \{f(x) \colon x \in [x_{k-1},x*]\}\\ t_2 &= \inf \{f(x) \colon x \in [x*, x_k]\} \end{align*}

Then t_1 \geq m_k and t_2 \geq m_k, where m_k = \inf \{f(x) \colon x \in [x_{k-1},x_k]\} as defined previously. All of the terms in L(f,P*) and L(f,P) are the same except for those over the interval [x_{k-1},x_k]. Thus we have

\begin{align*} L(f,P*) - L(f,P) &= [t_1(x* - x_{k-1}) + t_2(x_k - x*)] - [m_k(x_k - x_{k-1})]\\ &= (t_1 - m_k)(x* - x_{k-1}) + (t_2 - m_k)(x_k - x*). \end{align*}

This final sum is positive since all of the terms are positive. Thus L(f,P) \leq L(f,P*). If our partition Q contains r more points than P, we apply this argument r times to obtain L(f,P) \leq L(f,Q).

The proof of U(f,Q) \leq U(f,P) is similar.

Theorem 4.1.2

Let f be a bounded function on [a,b]. If P and Q are partitions of [a,b], then L(f,P) \leq U(f,Q).

Direct Proof

If P and Q are equal the proof is trivial, so we need only consider the case where P and Q are seperate partitions. Let R be the partition given by P \cup Q. R is a refinement of both P and Q. By Theorem 4.1.1 we have

\begin{align*} L(f,P) \leq L(f,R) &\leq U(f,R) \leq U(f,P)\\ L(f,Q) \leq L(f,R) &\leq U(f,R) \leq U(f,Q)\\ \end{align*}

Thus L(f,P) \leq U(f,Q) for all partitions P and Q.

Riemann Integral

Definition

Let f be a bounded function defined on [a,b] and P be a partition on [a,b]. Then

U(f) = \inf \{U(f,P)\} \qquad L(f) = \sup \{L(f,P)\}

are called the upper integral and lower integral of f on [a,b], respectively.

Definition

If L(f) = U(f), we say that f is Riemann integrable on [a,b], and notate the value with

\int^a_b f \quad \text{ or } \quad \int^a_b f(x) \, dx.

This is known as the Riemann integral of f on [a,b].

If a function is Riemann integrable, then the value of the integral on [a,b] will be equal to the signed area between the function and the x-axis.

Corollary 4.1.3

Let f be a bounded function on [a,b]. Then L(f) \leq U(f).

Direct Proof

If P and Q are partitions of [a,b], we have that L(f,P) \leq U(f,Q) by Theorem 4.1.2. Then it follows that U(f,Q) is an upper bound of the set defined by

S = \{L(f,P) \colon P \text{ partitions } [a,b]\}.

Thus U(f,Q) \geq \sup S. That is, L(f) \leq U(f,Q) for all partitions Q of [a,b]. Hence

L(f) \leq \inf \{U(f,Q) \colon Q \text{ partitions } [a,b]\} = U(f).

Essentially this Corollary states that every lower sum will be less than any upper sum.

Not every function is going to be Riemann integrable.

Example (Non-integrable functions)

Let f \colon [0,1] \to \mathbb{R} be defined

g(x) = \begin{cases} 1, & \text{if $x$ is rational}\\ 0, & \text{if $x$ is irrational.} \end{cases}

Let P = \{x_0,\ldots,x_n\} be any partition of [0,1]. Since each subinterval [x_{i-1},x_i] contains both rational and irrational points, we have M_i = 1 and m_i = 0 for all i \in \{1,\dots,n\}. Therefore

U(f,P) = \sum^n_{i=1}(1)\Delta x_i = 1 \qquad L(f,P) = \sum^n_{i=1}(0)\Delta x_i = 0.

Since the lower and upper integrals are different for every partition P, f is non-integrable on [0,1]. This special function f is known as the Dirichlet function.

Theorem 4.1.4 (Riemann integrablility Criterion)

Let f be a bounded function on [a,b]. Then f is integrable if and only if for each \varepsilon > 0 there exists a partition P of [a,b] such that

U(f,P) - L(f,P) < \varepsilon.

Direct Proof

Forward direction

Suppose f is integrable, meaning L(f) = U(f). Given any \varepsilon > 0, there exists a partition P_1 of [a,b] such that

L(f,P_1) > L(f) - \frac{\varepsilon}{2}.

This follows since L(f) is a supremum. Similarly, there exists a partition P_2 of [a,b] such that

U(f,P_2) < U(f) + \frac{\varepsilon}{2}.

Let P = P_1 \cup P_2. Then by Theorem 4.1.1 we have

\begin{align*} U(f,P) - L(f,P) &\leq U(f,P_2) - L(f,P_1)\\ &< \left[U(f) + \frac{\varepsilon}{2}\right] - \left[L(f) - \frac{\varepsilon}{2}\right]\\ &= U(f) - L(f) + \varepsilon = \varepsilon. \end{align*}

Therefore f being integrable implies there exists P such that U(f,P) - L(f,P) < \varepsilon.

Reverse direction

Now, suppose we have P such that U(f,P) - L(f,P) < \varepsilon. Then

U(f) \leq U(f,P) < L(f,P) + \varepsilon \leq L(f) + \varepsilon.

Since \varepsilon > 0 is arbitrary, we have U(f) \leq L(f). By Theorem 4.1.3 we have that L(f) \leq U(f). Thus L(f) = U(f), and f is Riemann integrable.