The Riemann Integral

Let $f$ be a bounded function defined on the closed and bounded interval $[a, b]$. Let $P = \{ a = x_0, x_1, ..., x_n = b \}$ be a partition of $[a, b]$ with:

(1)

\begin{align} \quad a = x_0 < x_1 < ... < x_n = b \end{align}

Let $\mathcal P[a, b]$ denote the set of all partitions on $[a, b]$. For each $i \in \{ 1, 2, ..., n \}$ we define:

(2)

\begin{align} \quad M_i &= \sup \{ f(x) : x \in [x_{i-1}, x_i] \} \\ \quad m_i &= \inf \{ f(x) : x \in [x_{i-1}, x_i] \} \end{align}

With the notation above we can define the upper and lower Riemann sums associated with the partition $P$ for the function $f$.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a, b]$ and let $P \in \mathcal \wp [a, b]$. The Upper Riemann Sum Associated with the Partition $P$ for $f$ is $\displaystyle{U(P, f) = \sum_{i=1}^{n} M_i \Delta x_i}$. The Lower Riemann Sum Associated with the Partition $P$ for $f$ is $\displaystyle{L(P, f) = \sum_{i=1}^{n} m_i \Delta x_i}$.

It can be shown that for any partitions $P_1, P_2 \in \mathcal \wp [a, b]$ with $P_1 \subseteq P_2$ we have that:

(3)

\begin{align} \quad U(P_2, f) \leq U(P_1, f) \quad \mathrm{and} \quad L(P_1, f) \leq L(P_2, f) \end{align}

So as we consider partitions that are finer and finer, $U(P, f)$ decreases and $L(P, f)$ increases. It can also be shown that for any partitions $P, P' \in \wp [a, b]$:

(4)

\begin{align} \quad L(P, f) \leq U(P', f) \end{align}

That is, the set of all upper Riemann sums is bounded below by any lower Riemann sum, and the set of all lower Riemann sums is bounded above by any upper Riemann sum. We now define the upper and lower Riemann integrals of a bounded function $f$ on $[a, b]$.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a, b]$. The Upper Riemann Integral of $f$ is defined to be $\displaystyle{(R) \overline{\int_a^b} f(x) \: dx = \inf \{ U(P, f) : P \in \wp [a, b] \}}$ and the Lower Riemann Integral of $f$ is defined to be $\displaystyle{(R) \underline{\int_a^b} f(x) \: dx = \sup \{ L(P, f) : P \in \wp [a, b] \}}$.

Another way t o define the upper and lower Riemann integrals of $f$ is through step functions. If $\varphi$ is a step function defined on $[a, b]$ then it is easy to show that the upper and lower Riemann integrals of $\varphi$ exist and define the upper and lower Riemann integrals of $f$ to also be:

(5)

\begin{align} \quad (R) \overline{\int_a^b} f(x) \: dx = \inf \left \{ (R) \int_a^b \psi(x) \: dx : \psi \: \mathrm{is \: a \: step \: function}, \: f(x) \leq \psi (x) \: \mathrm{on \:} [a, b] \right \} \end{align}

(6)

\begin{align} \quad (R) \underline{\int_a^b} f(x) \: dx = \sup \left \{ (R) \int_a^b \varphi(x) \: dx : \varphi \: \mathrm{is \: a \: step \: function}, \: \varphi(x) \leq f(x) \: \mathrm{on \:} [a, b] \right \} \end{align}

We are finally able to define what it means for a bounded function $f$ defined on a closed and bounded interval $[a,b]$ to be Riemann integrable.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a, b]$. Then $f$ is said to be Riemann Integrable on $[a, b]$ denoted $f \in R[a, b]$ if $\displaystyle{(R) \overline{\int_a^b} f(x) \: dx = \underline{\int_a^b} f(x) \: dx}$.